{ "paper_info": { "title": "Effective version of Ratner's equidistribution theorem for SL(3,R)", "total_elements": 39, "elements_with_proofs": 38, "counting_convention": "Counts theorem, lemma, proposition and corollary statements, excluding definitions and remarks. The proof count includes the equivalence argument for thm:effective-U-orbit and the assembled proof of prop:improve-dimension. The contextual Ratner theorem has no supplied proof.", "historical_review": { "cutoff_date": "2022-09-26", "basis": "journal_received", "date_evidence": "Annals publisher page https://annals.math.princeton.edu/2025/202-1/p03, Milestones: Received 26 September 2022; cached publisher.html.", "inspection_limit": "The cutoff record was supplied with the review instructions. No external reference or publisher page was independently inspected in this review.", "manuscript_version": "The supplied text has no dated version identifier and contains bibliography entries from 2023 and 2024. Its relationship to the received version is unresolved.", "policy_application": "No post-cutoff theorem is admitted as justification. In particular, the cited 2023 journal version of LM-preprint2021 does not establish availability of Proposition 4.2 by the cutoff. A dated preprint version containing that proposition is required.", "knowledge_limit": "Model weights cannot exclude later mathematical knowledge. The assessments below rely on direct checks of the supplied argument, and external results are left unaudited rather than certified from memory.", "requested_records": [ "A dated manuscript version corresponding to the received submission.", "A pre-cutoff version of LM-preprint2021 containing the Proposition 4.2 actually invoked.", "Exact locators and dated statements for the effective closing arguments attributed to einsiedler-margulis-venkatesh." ] } }, "verification_results": [ { "label": "Theorem 1.1", "element_type": "theorem", "element_id": "theorem_ratner", "source_label": "thm:ratner", "argument_gist": "States the qualitative orbit-closure and equidistribution theorem to explain the effective problem. It is not used as a proved prerequisite of the effective argument.", "dependencies": [], "external_inputs": [], "external_references": [ { "bibliography_key": "ratner_2", "role": "background", "locator": "", "audit_status": "not_inspected", "evidence": "In 1991, Ratner \\cite{ratner_2} proved the following fundamental theorem on" } ], "verdict": "unverifiable", "issues": [ "No proof is supplied, and the external theorem was not inspected. This is a contextual statement, not an identified gap in the main proof." ], "reasoning": "The formulation is recognizable as the qualitative theorem, subject to the usual meaning of an Ad-unipotent flow for a general Lie group. Its inclusion does not certify the subsequent effective estimates." }, { "label": "Theorem 1.2", "element_type": "theorem", "element_id": "theorem_effective_u", "source_label": "thm:effective-U-orbit", "argument_gist": "Transfers the expanding-segment formulation to a long unipotent segment using a(t)u(r)=u(e^t r)a(t) and the commutation of a(t) with a_0(ell).", "dependencies": [ { "source_label": "thm:main-thm", "evidence": "Proving Theorem \\ref{thm:effective-U-orbit} is equivalent to proving the following theorem:" } ], "external_inputs": [], "external_references": [], "verdict": "major_issues", "issues": [ "The equivalence is correct, but the proof of thm:main-thm is not established; see its entry." ], "reasoning": "Substituting y_0=a(t)x_0 correctly transforms both alternatives. There is no additional independent proof of the effective assertion." }, { "label": "Theorem 1.3", "element_type": "theorem", "element_id": "theorem_main", "source_label": "thm:main-thm", "argument_gist": "Removes exceptional orbit portions, obtains initial transverse dimension, repeatedly improves that dimension through weighted Kakeya incidence estimates, and converts near-maximal transverse dimension into effective equidistribution.", "dependencies": [ { "source_label": "prop:initial-bound-dimension", "evidence": "By Proposition \\ref{prop:initial-bound-dimension}, we have that $\\mu_{\\cF_0}$ is $(d_0, s_0)$-good." }, { "source_label": "prop:improve-dimension", "evidence": "By Proposition \\ref{prop:dimension-improvement-random}, \\ref{prop:dimension-improvement-strange-part}, and \\ref{prop:dimension-improvement-structured-component} we get Proposition \\ref{prop:improve-dimension}." }, { "source_label": "prop:reduction", "evidence": "By Proposition \\ref{prop:reduction}, it implies that $\\mu_{\\cF_{i}}$ is $(d_i, s_i)$-good, where $d_i = d_0 + i \\epsilon_2$ and $s_i = 2^{-i}s_0$(we remove an exponentially small proportion at each step)." }, { "source_label": "prop:high-dim-decomposition", "evidence": "By Proposition \\ref{prop:high-dim-decomposition}, we have that" }, { "source_label": "prop:high-dimension-to-equidistribution-a", "evidence": "Then Proposition \\ref{prop:high-dimension-to-equidistribution-a}, we have \n that $a(2 s_i)_\\ast \\mu_{k}$ is effectively equidistributed." } ], "external_inputs": [], "external_references": [], "verdict": "major_issues", "issues": [ "The foundational prop:quantitative-nondivergence is false as stated.", "The initial dimension and subgroup non-concentration arguments contain substantive gaps and, in prop:sl2xr2-closing-lemma, a reversed conjugation.", "The dimension-improvement chain uses incorrect conditioning inequalities and an unproved incidence geometry package; lm:hyperbola-plane contains a false quantitative implication.", "The final prop:high-dim-decomposition claims an exact positive measure decomposition but proves, at most, a weak approximation.", "The transfer of the external equidistribution estimate in prop:high-dimension-to-equidistribution-a remains unverified, including its pre-cutoff source version." ], "reasoning": "The terminal bookkeeping identity is consistent: the accumulated diagonal time before the final translate is t-2s_i. The problem is not that identity but the failure to establish several indispensable inputs. These objections do not constitute a counterexample to the main theorem itself." }, { "label": "Corollary 1.4", "element_type": "corollary", "element_id": "corollary_straight_line", "source_label": "cor:straight-line", "argument_gist": "Conjugates the expanding line into the main theorem's U-segment form and intends to rule out the exceptional alternative by the simultaneous Diophantine hypothesis on w.", "dependencies": [ { "source_label": "thm:main-thm", "evidence": "Applying Theorem \\ref{thm:main-thm} with $x_0 = a_0(t) n^\\ast(-w_1, w_2)\\Gamma$ and $f_{w_1}$, we have that" } ], "external_inputs": [], "external_references": [ { "bibliography_key": "KNSY", "role": "background", "locator": "", "audit_status": "not_inspected", "evidence": "The qualitative version of Corollary \\ref{cor:straight-line} was proved in \\cite{KNSY} for $\\omega(\\vv w) \\le 2$ using Ratner's theorem." } ], "verdict": "major_issues", "issues": [ "The main theorem is not established.", "The assertion that the Diophantine condition excludes every exceptional ell is not proved. Both the standard and exterior-square representations must be checked; the standard representation involves a dual linear-form estimate, not merely the displayed simultaneous lower bound.", "The stated definition uses a uniform lower bound at the endpoint exponent. This is stronger than merely specifying the usual supremal Diophantine exponent and should be distinguished terminologically." ], "reasoning": "The displayed matrix factorization and preservation of the integral under z(w_1) are correct. The remaining Diophantine assertion is plausible under the explicitly stated uniform hypothesis, but it is an omitted proof obligation rather than a verified step." }, { "label": "Corollary 1.6", "element_type": "corollary", "element_id": "corollary_curve", "source_label": "cor:curve", "argument_gist": "Approximates short curve pieces by tangent U-segments, uses quantitative non-divergence to discard bad centers, and applies the main theorem to the remaining pieces.", "dependencies": [ { "source_label": "thm:main-thm", "evidence": "By Theorem \\ref{thm:main-thm}, if we have that for any" } ], "external_inputs": [ { "bibliography_key": "Bernik-Kleinbock-Margulis", "role": "proof_input", "imported_result": "Theorem 1.4, invoked to obtain a uniform exponentially small bound for Leb(fm_ell). The exact external statement is not supplied.", "required_hypotheses": [ "The parameter-dependent lattice map must satisfy the theorem's quantitative goodness and nondegeneracy hypotheses.", "The required lower bounds on the relevant covolume functions must hold uniformly over the stated ell range.", "The theorem must control the compactness condition involving both vectors and exterior-square vectors." ], "locator": "Theorem 1.4", "audit_status": "not_inspected", "available_from": "2001", "date_evidence": "\\newblock {\\em International Mathematics Research Notices}, 2001(9):453--486,\n 2001.", "date_basis": "Publication year reported by the manuscript bibliography; external date and theorem not independently inspected.", "version": "Cited 2001 journal version; exact theorem hypotheses unverified.", "evidence": "By \\cite[Theorem 1.4]{Bernik-Kleinbock-Margulis}, we have that" } ], "external_references": [ { "bibliography_key": "Shah_2", "role": "background", "locator": "", "audit_status": "not_inspected", "evidence": "The ineffective version of Corollary \\ref{cor:curve} was proved in \\cite{Shah_2} using Ratner's theorem." } ], "verdict": "major_issues", "issues": [ "The claimed exponentially small tangent approximation at interval length e^{-t/2} is incorrect. For psi(r)=(r,r^2), writing r=r_0+e^{-t/2}v leaves a quadratic displacement v^2 in the E_13 coordinate after conjugation by a_1(t); it is of order one, not exponentially small.", "A general nondegenerate smooth curve is replaced by (r,psi_2(r)) without a local-chart decomposition, a treatment of the resulting density, or justification at points where the chosen coordinate derivative vanishes.", "The uniform exceptional-set estimate is asserted without checking the imported theorem's hypotheses.", "The bound for the union over all real ell does not follow from the bounds for individual ell. A discretization with an explicit compactness buffer is required.", "The proof changes [1] to [0,1] and does not justify how the good-center measure estimate yields a suitable partition into good short pieces." ], "reasoning": "The tangent factorization itself has the intended linear term. Its Taylor remainder has exactly the wrong size for the claimed approximation. This is a demonstrated error in the supplied corollary proof, independently of the issues with the main theorem." }, { "label": "Lemma 2.1", "element_type": "lemma", "element_id": "lemma_lie_bracket", "source_label": "lm:lie-bracket", "argument_gist": "Computes brackets between the explicitly listed matrix units and diagonal generators.", "dependencies": [], "external_inputs": [], "external_references": [], "verdict": "valid", "issues": [], "reasoning": "The inclusions and the three sl(2) identities follow directly from [E_ij,E_kl]=delta_jk E_il-delta_li E_kj. In particular, the transverse decomposition has the asserted bracket parity." }, { "label": "Lemma 2.2", "element_type": "lemma", "element_id": "lemma_product_lie_algebra", "source_label": "lm:product-lie-algebra-1", "argument_gist": "Uses the local BCH series and the transverse/block-diagonal bracket decomposition, then inverts the local product coordinate map.", "dependencies": [ { "source_label": "lm:lie-bracket", "evidence": "By Lemma \\ref{lm:lie-bracket} and the fact that for any $\\vv w_1, \\vv w_2 \\in \\fg$," } ], "external_inputs": [], "external_references": [], "verdict": "minor_issues", "issues": [ "A Lie algebra norm with the claimed bracket bound has not been specified.", "The local product-coordinate existence in the converse should be justified by the inverse function theorem.", "The last constant 4 does not follow from merely substituting the displayed coarse bounds ||w||<=2||tilde w|| and ||h||<=2||tilde h|| into the preceding estimate; tighter small-neighborhood bounds are needed." ], "reasoning": "The local statement is supported by the BCH expansion and the invertibility of the derivative of the product map. The quantitative constants need a fixed norm and slightly more careful bookkeeping; no structural obstruction was found." }, { "label": "Lemma 2.3", "element_type": "lemma", "element_id": "lemma_product_w", "source_label": "lm:product-w", "argument_gist": "Separates the even and odd transverse-degree terms in BCH and uses the preceding product-coordinate lemma.", "dependencies": [ { "source_label": "lm:lie-bracket", "evidence": "By Lemma \\ref{lm:lie-bracket}, we have the following observations:" }, { "source_label": "lm:product-lie-algebra-1", "evidence": "By Lemma \\ref{lm:product-lie-algebra-1}, we have that" } ], "external_inputs": [], "external_references": [], "verdict": "minor_issues", "issues": [ "The BCH tail bounds and numerical constants depend on an unstated norm convention.", "The application of lm:product-lie-algebra-1 uses a radius up to 3rho_0, although that lemma is stated only for radii below 10^{-10}; the smallness threshold should be adjusted." ], "reasoning": "The bracket parity is correct, and the orders O(rho rho_0) and O(rho) follow locally. The issues concern the literal thresholds and constants." }, { "label": "Proposition 3.1", "element_type": "proposition", "element_id": "proposition_high_dimension_equidistribution", "source_label": "prop:high-dimension-to-equidistribution-a", "argument_gist": "Proposes a van der Corput and mixing argument converting a nearly two-dimensional distribution in the expanding transverse plane into equidistribution.", "dependencies": [], "external_inputs": [ { "bibliography_key": "LM-preprint2021", "role": "proof_input", "imported_result": "Proposition 4.2 and its proof, claimed to transfer step by step to the present SL(3,R) action. The exact external theorem and the transfer are not supplied.", "required_hypotheses": [ "A suitable quantitative matrix-coefficient estimate for the present G action and Sobolev norm.", "The stated Frostman bound at scale beta e^{-s'}.", "Uniform control for base points in X_{beta^{1/4}}.", "Verification that the transverse directions, flow relations and parameter ranges in the cited proof apply here." ], "locator": "Proposition 4.2", "audit_status": "not_inspected", "available_from": null, "date_evidence": "\\newblock {\\em Inventiones Mathematicae}, 231(3):1141--1237, 2023.", "date_basis": "The cited journal version is post-cutoff. The bibliography key alone does not establish the date or contents of an earlier preprint.", "version": "Cited 2023 journal version; required pre-cutoff preprint version unidentified.", "evidence": "The statement can be proved by following the proof of \\cite[Proposition 4.2]{LM-preprint2021} step by step." }, { "bibliography_key": "Venkatesh2010", "role": "proof_input", "imported_result": "The van der Corput-type equidistribution argument in Section 3, cited as the method underlying this proposition. No exact transferable theorem is identified.", "required_hypotheses": [ "Quantitative mixing and the regularity bounds needed by the averaging argument.", "Control of correlations for the particular transverse measure and unipotent subgroup used here." ], "locator": "Section 3", "audit_status": "not_inspected", "available_from": "2010", "date_evidence": "\\newblock {\\em Annals of Mathematics}, 172:989--1094, 2010.", "date_basis": "Publication year reported by the manuscript bibliography; external version not independently inspected.", "version": "Cited 2010 journal version; exact imported argument unverified.", "evidence": "It can be proved by following Venkatesh's argument (\\cite[\\S 3]{Venkatesh2010} and \\cite[Proposition 4.2]{LM-preprint2021} for details)." } ], "external_references": [], "verdict": "unverifiable", "issues": [ "There is no supplied proof beyond a reference to another proof in a different homogeneous-space setting.", "The necessary adaptation, constants, Sobolev norm and quantitative mixing input are not identified.", "Availability of the needed LM Proposition 4.2 before 2022-09-26 is unresolved." ], "reasoning": "This is a central proof obligation, not a routine citation that can be certified without inspecting and adapting the source. No counterexample to the proposition is established here." }, { "label": "Proposition 4.1", "element_type": "proposition", "element_id": "proposition_quantitative_nondivergence", "source_label": "prop:quantitative-nondivergence", "argument_gist": "Intends to use quantitative non-divergence to reduce cusp escape to a uniformly short vector or bivector, then detect that obstruction at an a_0 endpoint.", "dependencies": [], "external_inputs": [ { "bibliography_key": "kleinbock2008", "role": "proof_input", "imported_result": "Theorem 2.2, invoked as quantitative non-divergence for the polynomial lattice map r -> a(s)u(r)g.", "required_hypotheses": [ "Uniform goodness of the relevant exterior-power covolume functions.", "A positive uniform lower bound for their suprema on the parameter interval.", "The theorem's lattice-rank and interval hypotheses." ], "locator": "Theorem 2.2", "audit_status": "not_inspected", "available_from": "2008", "date_evidence": "\\newblock {\\em Transactions of the American Mathematical Society},\n 360(12):6497--6523, 2008.", "date_basis": "Publication year reported by the manuscript bibliography; source not independently inspected.", "version": "Cited 2008 journal version.", "evidence": "By the Kleinbock-Margulis quantitative non-divergence theorem \n (cf. \\cite[Theorem 2.2]{kleinbock2008}, \\cite[Theorem 5.2]{Klein_Mar}), if the statement does not hold then there exists" }, { "bibliography_key": "Klein_Mar", "role": "proof_input", "imported_result": "Theorem 5.2, cited as an alternative quantitative non-divergence input.", "required_hypotheses": [ "Goodness of all relevant primitive-subgroup covolume functions.", "Uniform lower bounds on their suprema.", "The applicable interval and lattice hypotheses." ], "locator": "Theorem 5.2", "audit_status": "not_inspected", "available_from": "1998", "date_evidence": "\\newblock {\\em Annals of Mathematics}, pages 339--360, 1998.", "date_basis": "Publication year reported by the manuscript bibliography; source not independently inspected.", "version": "Cited 1998 journal version.", "evidence": "By the Kleinbock-Margulis quantitative non-divergence theorem \n (cf. \\cite[Theorem 2.2]{kleinbock2008}, \\cite[Theorem 5.2]{Klein_Mar}), if the statement does not hold then there exists" } ], "external_references": [], "verdict": "major_issues", "issues": [ "The proposition is false: its hypothesis says that both a_0 endpoints are outside the compact set, whereas the intended contradiction requires endpoint non-escape.", "Explicit counterexample: fix s=1, let g=diag(e^{-M},e^{M/2},e^{M/2}), and x=gGamma. For M>2/3 both endpoint lattices have e_1 shorter than e^{-1/3}. For any beta>e^{-M}, every a(1)u(r)x is outside X_beta, so the bad set has measure 1. Letting beta tend to zero and choosing M accordingly contradicts any uniform C beta^alpha bound.", "The proof's estimates also lose constants: on [-1/2,1/2], a linear expression bounded by 1 yields |v'_3|<=2e^{-s}, not the asserted e^{-s}. The final norm estimate does not follow from componentwise bounds with constant 1." ], "reasoning": "The counterexample uses a vector fixed by U and a(s), so it is independent of any uncertainty about the external theorem. Replacing the endpoint signs is an intended correction, not a verified repair of all quantitative details." }, { "label": "Lemma 4.2", "element_type": "lemma", "element_id": "lemma_separation", "source_label": "lm:separation-lemma-a", "argument_gist": "Conjugates an apparent near-return by a(-ell) and intends to contradict the injectivity radius in the thick part.", "dependencies": [], "external_inputs": [], "external_references": [], "verdict": "major_issues", "issues": [ "The lemma is false for the full stated range ell>=1. With the canonical coordinate norm, take ell=1, x=Gamma, r_1=beta/4, r_2=r_3=0 and v=(e beta/4)fu. For sufficiently small beta, a(-1)Gamma lies in X_{beta^{1/4}}, all hypotheses hold, and u*=v_2=w_2=0 violates the conclusion.", "The proof itself invokes ell sufficiently large to ensure gamma is nontrivial, a condition absent from the statement.", "The asserted membership in B^G(beta) requires constant control on a product of two beta-sized factors; it does not follow literally from the displayed estimates." ], "reasoning": "The counterexample is an equality already in the group, with gamma equal to the identity. A sufficiently-large-ell formulation may avoid this example, but that would be a changed statement." }, { "label": "Proposition 6.2", "element_type": "proposition", "element_id": "proposition_initial_dimension", "source_label": "prop:initial-bound-dimension", "argument_gist": "Turns excessive local concentration into many short near-returns, extracts integer matrices almost stabilizing the H-volume vector, and seeks a nearby exact stabilizer producing a short lattice vector in the lower plane.", "dependencies": [ { "source_label": "lm:separation-lemma-a", "evidence": "by Lemma \\ref{lm:separation-lemma-a}, if we write $h_{\\bar x} = \\exp(\\vv h_{\\bar x})$, and" } ], "external_inputs": [ { "bibliography_key": "einsiedler-margulis-venkatesh", "role": "proof_input", "imported_result": "An effective approximate-stabilizer-to-exact-stabilizer argument, used first for one integer matrix and then simultaneously for two noncommuting matrices. No theorem or lemma locator is supplied.", "required_hypotheses": [ "A specified representation and algebraic stabilizer.", "Height bounds for the integer matrices and the representative g.", "A quantitative distance-to-exact-stabilizer estimate with the stated exponents.", "For Case 2, a simultaneous estimate for both matrices." ], "locator": "", "audit_status": "not_inspected", "available_from": "2009", "date_evidence": "\\newblock {\\em Inventiones mathematicae}, 177(1):137--212, 2009.", "date_basis": "Publication year reported by the manuscript bibliography; the particular imported result is unidentified.", "version": "Cited 2009 journal version; exact closing result unresolved.", "evidence": "By repeating the same argument as in \\cite{einsiedler-margulis-venkatesh}," } ], "external_references": [], "verdict": "major_issues", "issues": [ "The construction of F and its asserted small discarded mass depend on the false prop:quantitative-nondivergence.", "The packing argument producing the stated number of separated near-returns, including a lower bound on their return times, is not supplied.", "Equation eq:a0ge bounds the coefficient a_0 of fa, but the subsequent distinctness argument invokes it as a lower bound on a_2, the fu* coefficient. The cited separation lemma would concern the latter coefficient, not the former.", "The assertion that an element of SL(3,Z) with nontrivial unipotent part must itself be unipotent is false. The matrix with rows (-1,1,0), (0,-1,0), (0,0,1) is a counterexample.", "The claimed stabilizer Stab(v_H)=A_0H omits a disconnected component: diag(-1,-1,1) fixes the H-volume vector but is not in A_0H.", "The effective closing estimates, especially simultaneous closing in Case 2, are not justified by an identified external result.", "Bounds for a bounded representative g and losses under Ad(g^{-1}) are used without being established at the point where g is chosen." ], "reasoning": "Some local mistakes could be corrected separately, for example by changing a coefficient or taking a power of an integer matrix. Such changes do not supply the missing packing and effective closing arguments. The proposition is therefore unproved here, not disproved by the auxiliary matrix examples." }, { "label": "Proposition 6.3", "element_type": "proposition", "element_id": "proposition_subgroup_nonconcentration", "source_label": "prop:sl2xr2-closing-lemma", "argument_gist": "Attempts to turn concentration near one transverse summand into concentration near both summands by applying a central diagonal translate, then repeats the initial closing argument.", "dependencies": [ { "source_label": "prop:initial-bound-dimension", "evidence": "By repeating the argument in the proof of \n Proposition \\ref{prop:initial-bound-dimension}, we can show that" } ], "external_inputs": [], "external_references": [], "verdict": "major_issues", "issues": [ "The displayed conjugation identity has the wrong sign. On fr_1, Ad(a_0(ell)) contracts by e^{-ell/2}; on fr_2 it expands by e^{ell/2}. Thus Q(fr_1,ell,beta) is sent to transverse sizes beta e^{-3ell/2} and beta e^{ell/2}, not beta e^{-ell/2} in both summands.", "The proposed repeated proof inherits the unresolved initial closing argument and exceptional-set construction.", "The proof must also verify all thick-part conditions for the translated measure, not only its center." ], "reasoning": "For j=1 the equalizing conjugation would have the opposite sign. This is a demonstrated error in the supplied reduction; a corrected conjugation would still require a complete proof of the preceding proposition." }, { "label": "Proposition 7.2", "element_type": "proposition", "element_id": "proposition_reduction", "source_label": "prop:reduction", "argument_gist": "Covers the measure by full playgrounds, discards light playgrounds and local exceptional sets, and absorbs a covering multiplicity into the dimension-improvement margin.", "dependencies": [], "external_inputs": [], "external_references": [], "verdict": "major_issues", "issues": [ "The first asserted finite cover of all X by at most beta^{-9} translates of Pi_{s'} is impossible. Pi_{s'} is compact for fixed parameters, so a finite union of its translates cannot cover the noncompact space X.", "A cover of an appropriate thick support might replace this claim, but its cardinality and the choice of centers must be proved.", "The proof switches between restricted measures and renormalized probability measures. Removing further mass does not preserve an exact probability upper bound without a normalization loss." ], "reasoning": "The union-bound idea is sound on a suitably controlled compact support with slack. The actual cover asserted in the proof is false and the required replacement is not supplied." }, { "label": "Lemma 8.1", "element_type": "lemma", "element_id": "lemma_two_theta", "source_label": "lm:two-theta", "argument_gist": "Combines the two transverse exponentials by BCH and moves the small block-diagonal error into the remaining factors.", "dependencies": [ { "source_label": "lm:product-w", "evidence": "By Lemma \\ref{lm:product-w}," } ], "external_inputs": [], "external_references": [], "verdict": "minor_issues", "issues": [ "The numerical radius 1.1 requires a coordinate-compatible norm and explicit control of the rearrangement.", "For a Euclidean norm, two orthogonal transverse vectors each at their allowed radius need not have sum of norm at most 1.01 times that radius. The intended box-norm convention should be stated." ], "reasoning": "The claimed scale orders follow from the bracket calculations. The literal constants are not justified for an arbitrary norm." }, { "label": "Lemma 8.2", "element_type": "lemma", "element_id": "lemma_product_theta", "source_label": "lm:product-theta", "argument_gist": "Uses the enlarged Theta coordinate description, combines transverse factors, and controls the stable block-diagonal coordinate under products and inverses.", "dependencies": [ { "source_label": "lm:two-theta", "evidence": "By Lemma \\ref{lm:two-theta}, it suffices to show that" }, { "source_label": "lm:product-w", "evidence": "By Lemma \\ref{lm:product-w}, we have" } ], "external_inputs": [], "external_references": [], "verdict": "minor_issues", "issues": [ "The anisotropic estimate for the fu* coordinate after rearrangement is left implicit.", "The precise constant 2.21 depends on the unresolved norm convention in lm:two-theta." ], "reasoning": "The small-scale multiplication mechanism is consistent with the Lie brackets. The omitted estimates appear local and repairable, unlike the later global conditioning and incidence steps." }, { "label": "Corollary 8.3", "element_type": "corollary", "element_id": "corollary_product_theta", "source_label": "cor:product-theta", "argument_gist": "Applies the two-factor neighborhood estimate twice to obtain a four-factor estimate.", "dependencies": [ { "source_label": "lm:product-theta", "evidence": "By repeating the proof of Lemma \\ref{lm:product-theta}, we get" } ], "external_inputs": [], "external_references": [], "verdict": "minor_issues", "issues": [ "Inherits the norm and numerical-radius qualifications of lm:product-theta." ], "reasoning": "Once a uniform two-factor estimate at the enlarged radius is proved, the stated iteration follows." }, { "label": "Lemma 8.4", "element_type": "lemma", "element_id": "lemma_different_coordinates", "source_label": "lm:different-coordinate-1", "argument_gist": "Uses local product coordinates and BCH to show that changing the order of transverse and block-diagonal factors introduces quadratic diagonal and cubic transverse errors.", "dependencies": [ { "source_label": "lm:product-w", "evidence": "by Lemma \\ref{lm:product-w}, we have" } ], "external_inputs": [], "external_references": [], "verdict": "minor_issues", "issues": [ "The initial product-coordinate existence and uniqueness are assumed without an inverse-function argument.", "The exact small constants need a fixed norm and a sufficiently small beta.", "The lemma should explicitly specify the intended nonnegative range of ell and the smallness assumptions on beta." ], "reasoning": "The quadratic and cubic orders are supported by the symmetric transverse/block-diagonal bracket decomposition. The remaining concerns are local quantitative details." }, { "label": "Lemma 8.5", "element_type": "lemma", "element_id": "lemma_cross_section", "source_label": "lm:cross-section", "argument_gist": "Uses the explicit orbit-coordinate formulas to solve for a prescribed U-coordinate and to estimate the time spent in a cross-section.", "dependencies": [ { "source_label": "eq:fy", "evidence": "Moreover, by \\eqref{eq:fy}, we have" } ], "external_inputs": [], "external_references": [], "verdict": "major_issues", "issues": [ "Solving f_y(tilde r)=0 controls only the projected U-coordinate. The proof does not show that the other coordinates remain within the defining bounds of Sigma^r.", "The claim Sigma(u(r)x) subset Pi(x) is used without boundary margins; translating a cross-section at an endpoint can exceed the playground's U-range.", "The reduction to r=0 and the sign-change argument must preserve the permitted time interval. This is not checked.", "The lower interval inclusion requires the transverse and diagonal coordinates to remain admissible throughout that interval, not merely |f_z(r)|<=beta." ], "reasoning": "The projected time coordinate has the expected derivative near one. That fact alone does not establish membership in the full anisotropic neighborhood. This is an unverified geometric step, not a counterexample to a buffered formulation." }, { "label": "Lemma 8.6", "element_type": "lemma", "element_id": "lemma_initial_dimension_after_translate", "source_label": "lm:initial-dimension-under-a-t", "argument_gist": "Pulls an anisotropic cross-section back by a(-2s'), applies initial dimension control, and uses its short U-width for an additional e^{-2s'} factor.", "dependencies": [ { "source_label": "lm:different-coordinate-1", "evidence": "By Lemma \\ref{lm:different-coordinate-1}, we have that" }, { "source_label": "prop:initial-bound-dimension", "evidence": "By repeating the proof of Proposition \\ref{prop:initial-bound-dimension} to $\\cF_i$ we have" } ], "external_inputs": [], "external_references": [], "verdict": "major_issues", "issues": [ "The conditioning inequality is incorrect. For A contained in a full playground, tilde mu(A)=mu(A)/mu(Pi(x)); one cannot bound it by the unconditioned pulled-back measure without the factor mu(Pi(x))^{-1}, potentially beta^{-15}.", "The measure has been restricted by exceptional-set removal; being a restriction of orbit Lebesgue measure does not automatically make it uniform along complete U-segments.", "The proof uses initial dimension control at radius 2beta and arbitrary centers, beyond the literal statement proved for radius beta and centers in F." ], "reasoning": "The missing conditioning denominator is an explicit measure-theoretic error. The later strange-part proof itself reintroduces a beta^{-15} factor, showing the distinction matters." }, { "label": "Lemma 8.8", "element_type": "lemma", "element_id": "lemma_partition_one", "source_label": "lm:partition-1", "argument_gist": "Rounds the three projected coordinates to a grid and uses neighborhood product estimates to bound overlap multiplicity.", "dependencies": [ { "source_label": "lm:product-theta", "evidence": "By Lemma \\ref{lm:product-theta}, we have" }, { "source_label": "lm:different-coordinate-1", "evidence": "By Lemma \\ref{lm:different-coordinate-1}, we have" } ], "external_inputs": [], "external_references": [], "verdict": "major_issues", "issues": [ "The proof does not establish the global uniqueness of the playground coordinates used by the projection P.", "The inference from eq:z-z' to a small axis-aligned difference P(z)-P(z') ignores the Heisenberg cross term. A transverse increment of size beta e^{-s'} against a coordinate of size beta e^{s'} contributes a U-coordinate shear of order beta^2, not beta e^{-s'}. A sheared-box counting argument is needed.", "Grid endpoints and the use of balls versus the declared centered intervals are not consistently controlled." ], "reasoning": "A bounded-overlap cover may be obtainable in suitable sheared coordinates. The displayed coordinate-difference justification does not prove the asserted multiplicity." }, { "label": "Lemma 8.9", "element_type": "lemma", "element_id": "lemma_partition_two", "source_label": "lm:partition-2", "argument_gist": "Rounds the two contracting transverse coordinates on a finer grid and uses product estimates to obtain an Omega cover with bounded overlap.", "dependencies": [ { "source_label": "lm:two-theta", "evidence": "By Lemma \\ref{lm:two-theta}, $\\Theta_{s'} \\subset \\hat{\\Theta}_{s'}$." }, { "source_label": "lm:product-w", "evidence": "By Lemma \\ref{lm:product-w}, we have that" }, { "source_label": "lm:different-coordinate-1", "evidence": "By Lemma \\ref{lm:different-coordinate-1}," } ], "external_inputs": [], "external_references": [], "verdict": "major_issues", "issues": [ "The final equality between the coordinate difference D_z(z')-D_z(y) and the relative transverse coordinate requires a coordinate-composition calculation; it is not a consequence of the preceding factorization alone.", "The grid range, the assertion I_z subset Theta'(z), and boundary coverage are not reconciled with the enlarged radii used in the proof.", "The use of local coordinates on X inherits the unestablished injectivity assumptions in the playground construction." ], "reasoning": "The fine-scale covering idea is reasonable, but the bounded-overlap statement is indispensable for the weighted incidence sums and has not been rigorously derived." }, { "label": "Lemma 8.12", "element_type": "lemma", "element_id": "lemma_contracting_coordinates_flow", "source_label": "lm:wv2r", "argument_gist": "Computes the two contracting transverse coordinates under U using the ordered matrix factorization.", "dependencies": [], "external_inputs": [], "external_references": [], "verdict": "minor_issues", "issues": [ "The concluding comparison for Delta requires the relevant relative-coordinate chart to be defined; this qualification is missing for the stated points.", "Small beta is required to keep 2 plus or minus r w_2 v_2 bounded away from zero." ], "reasoning": "The rational formulas have the asserted scale behavior because |r w_2 v_2| is O(beta^3). Within the intended small injective chart, the comparability follows." }, { "label": "Lemma 8.13", "element_type": "lemma", "element_id": "lemma_omega_flow", "source_label": "lm:omega-u", "argument_gist": "Uses the shearing relation Ad(u(r))r^- subset r^-+r^+ and an H time change to transport Omega packets between cross-sections.", "dependencies": [ { "source_label": "lm:cross-section", "evidence": "By Lemma \\ref{lm:cross-section}, $u(\\fl(y'))y' \\cap \\Sigma^{\\tilde{r}}$ is contained in an interval of length $\\asymp \\beta$." } ], "external_inputs": [], "external_references": [], "verdict": "major_issues", "issues": [ "The set inclusion depends on unproved cross-section boundary control.", "The implication from packet transport to comparable tilde mu masses assumes uniform conditional orbit density and comparable complete orbit lengths after restriction; these hypotheses have not been established.", "The proof starts with the inverse relation for y in terms of y', although membership was defined in the opposite direction, and needs anisotropic inverse estimates with the stated constants.", "A time-change estimate for derivatives or Jacobians is needed for the mass comparison, not only proximity of the times." ], "reasoning": "The Lie algebra shear has the correct scales. The geometric and measure-theoretic conclusions require additional arguments." }, { "label": "Lemma 8.14", "element_type": "lemma", "element_id": "lemma_time_leave", "source_label": "lm:time-leave", "argument_gist": "Bounds how long two packets can remain in a common Theta neighborhood by comparing expanding transverse coordinates and their relative contracting slope.", "dependencies": [ { "source_label": "lm:omega-u", "evidence": "By Lemma \\ref{lm:omega-u}, we have $\\Omega(y_i, L_i) \\subset \\hat{\\Omega}(\\tilde{y}_i)$." }, { "source_label": "cor:product-theta", "evidence": "By Corollary \\ref{cor:product-theta}," }, { "source_label": "lm:different-coordinate-1", "evidence": "By Lemma \\ref{lm:different-coordinate-1}, if we write" } ], "external_inputs": [], "external_references": [], "verdict": "major_issues", "issues": [ "The proof treats hat Omega as contained in Theta, although hat Omega has radii 2.1beta and Theta has radius beta in shared coordinates. The resulting four-Theta containment and constant 4.6 do not follow.", "The conversion from increments based at the tilde y_i to the originally defined I(y_1,y_2) needs explicit time shifts and time-change control.", "The relative-coordinate and packet-transport prerequisites remain unproved." ], "reasoning": "The rational expanding-coordinate formulas support an upper residence-time estimate in a correctly buffered chart. The supplied neighborhood inclusions do not establish the stated version." }, { "label": "Lemma 8.15", "element_type": "lemma", "element_id": "lemma_concentrated_surface", "source_label": "lm:concentrated-surface", "argument_gist": "Eliminates the encounter time from two rational transverse-coordinate equations to place interacting packet coordinates near a hyperbola and inside a small box.", "dependencies": [ { "source_label": "lm:time-leave", "evidence": "By Lemma \\ref{lm:time-leave}, $ r \\in I(\\bar z, y)$ implies that $|f_v(r)|, |f_w(r)| \\le 4.6 \\beta e^{-s'}$." } ], "external_inputs": [], "external_references": [], "verdict": "major_issues", "issues": [ "The existence of the selected y', z' and nonzero v_1,w_1 with the required uniform lower bounds is asserted without proof.", "The division leading to eq:2 and eq:1 is unjustified when |v_1| or |w_1| is only beta e^{-s'}, while the permitted residual is 4.6beta e^{-s'}. That residual can be larger than the quantity being inverted.", "For example, v_2=0 and v_1=beta e^{-s'} satisfy the residual inequality, but the claimed multiplicative equation r^{-1}+Delta=-(v_2/v_1)(1+O(...)) would force r^{-1}=0.", "The final paragraph drops the beta in the assumed Delta bound and nevertheless concludes a bound with beta^2.", "The proof's conversion from a polynomial residual to geometric distance from the hyperbola requires a lower gradient estimate." ], "reasoning": "The formal zero-residual elimination does yield the displayed hyperbola. The tolerance estimates needed for the actual thickened incidence statement are not established." }, { "label": "Proposition 9.1", "element_type": "proposition", "element_id": "proposition_improve_dimension", "source_label": "prop:improve-dimension", "argument_gist": "Decomposes heavy neighborhoods into random, strange and structured classes and combines small-mass estimates for all three to improve transverse dimension.", "dependencies": [ { "source_label": "prop:dimension-improvement-random", "evidence": "By Proposition \\ref{prop:dimension-improvement-random}, \\ref{prop:dimension-improvement-strange-part}, and \\ref{prop:dimension-improvement-structured-component} we get Proposition \\ref{prop:improve-dimension}." }, { "source_label": "prop:dimension-improvement-strange-part", "evidence": "By Proposition \\ref{prop:dimension-improvement-random}, \\ref{prop:dimension-improvement-strange-part}, and \\ref{prop:dimension-improvement-structured-component} we get Proposition \\ref{prop:improve-dimension}." }, { "source_label": "prop:dimension-improvement-structured-component", "evidence": "By Proposition \\ref{prop:dimension-improvement-random}, \\ref{prop:dimension-improvement-strange-part}, and \\ref{prop:dimension-improvement-structured-component} we get Proposition \\ref{prop:improve-dimension}." } ], "external_inputs": [], "external_references": [], "verdict": "major_issues", "issues": [ "All three constituent estimates have unresolved prerequisites or proof gaps.", "The proposition omits the upper restriction on d_i required by prop:dimension-improvement-structured-component.", "The heavy-neighborhood argument does not justify the exact number of covering Theta packets or the asserted bound after renormalizing the complement.", "Definition def:structure-randomness assumes a packet bound without beta^{-15}, stronger than the preceding lm:omega-upper establishes." ], "reasoning": "The intended decomposition is a coherent strategy, but it is not a completed proof. The main theorem's intended stopping range can satisfy the structured proposition's restriction; that does not repair the unrestricted proposition as written." }, { "label": "Lemma 9.4", "element_type": "lemma", "element_id": "lemma_omega_upper", "source_label": "lm:omega-upper", "argument_gist": "Pulls Omega back to an isotropically thin transverse neighborhood and uses the short U-width and the previous dimension bound.", "dependencies": [], "external_inputs": [], "external_references": [], "verdict": "major_issues", "issues": [ "The (d_i,2s')-good hypothesis is at radius beta, but the proof applies it directly at radius 2beta without a covering loss.", "The e^{-2s'} factor requires uniformity along appropriate complete U-pieces, which is not ensured after the exceptional removals.", "The later def:structure-randomness silently discards this lemma's beta^{-15} conditioning factor." ], "reasoning": "The conjugated transverse scales are correct. The measure estimate requires an orbitwise disintegration and radius-enlargement argument not supplied here." }, { "label": "Proposition 9.9", "element_type": "proposition", "element_id": "proposition_random_dimension", "source_label": "prop:dimension-improvement-random", "argument_gist": "Obtains a lower weighted intersection count from recurrent visits to normal heavy neighborhoods, and an incompatible upper count from non-concentration near encounter hyperbolas.", "dependencies": [ { "source_label": "lm:time-leave", "evidence": "By \\eqref{eq:assumption-1} and Lemma \\ref{lm:time-leave}," }, { "source_label": "lm:concentrated-surface", "evidence": "By Lemma \\ref{lm:concentrated-surface}, $\\fD_{z}(z') \\in \\cC(\\cT(y), z)$." } ], "external_inputs": [], "external_references": [], "verdict": "major_issues", "issues": [ "The lower count relies on unproved packet mass transport, bounded overlap and residence-time estimates.", "The asserted beta-separated sample size is not guaranteed with the exact constants from the displayed length lower bound; a constant-factor loss is needed.", "The upper count uses the stronger packet-weight hypothesis introduced in def:structure-randomness, not the previously proved beta^{-15}-weaker estimate.", "The equality expressing the pair count as a sum over heavy neighborhood centers is not exact for overlapping covers; a bounded-multiplicity inequality must be proved.", "Several uses of mu_fr have no independently defined measure in this portion of the argument." ], "reasoning": "If the geometric and conditional-measure inputs were rigorously established with slack, the opposing exponents would express the intended contradiction. As supplied, those inputs and the counting identities are not justified." }, { "label": "Proposition 9.10", "element_type": "proposition", "element_id": "proposition_strange_dimension", "source_label": "prop:dimension-improvement-strange-part", "argument_gist": "Repeats the weighted incidence argument for packets concentrated at an intermediate contracting scale j, gaining an extra initial-dimension factor e^{-d_0 j}.", "dependencies": [ { "source_label": "prop:dimension-improvement-random", "evidence": "Using similar arguments as in the proof of Proposition \\ref{prop:dimension-improvement-random}, we can get" }, { "source_label": "lm:time-leave", "evidence": "Since $\\|\\Delta(y, y_2)\\| \\ge \\beta e^{-s' - j-1}$, by Lemma \\ref{lm:time-leave}, for each $y_2 \\in \\fJ\\cap \\Sigma^r$ such that $(y, y_2) \\in \\diamond_{r,j}$, we have that $I(y, y_2) \\subset I$ where $I$ is an interval of length $\\le e^{j+1}$." }, { "source_label": "lem:sigma j lemma", "evidence": "Using similar arguments to those used in the proof of Lemma~\\ref{lem:sigma j lemma}, we can prove that" }, { "source_label": "lm:initial-dimension-under-a-t", "evidence": "By Lemma \\ref{lm:initial-dimension-under-a-t}, we have" }, { "source_label": "lm:concentrated-surface", "evidence": "By Lemma \\ref{lm:concentrated-surface}, the coordinates $\\fD_{ z}(y_1)$ of all $y_1\\in \\cI_z$ such that $(y_2,y_1) \\in \\diamond_{r,j}$" } ], "external_inputs": [], "external_references": [], "verdict": "major_issues", "issues": [ "The proof initially establishes a visit length beta^{30}e^{2s'} but then asserts the stronger beta^{22}e^{2s'} lower bound. For beta<1 this does not follow.", "The precise annular Delta condition is not proved from membership in Xi_j minus Xi_{j+1}; changes of packet representatives and time changes introduce constants.", "The localization lemma and the initial translated dimension estimate used in the upper bound are not established.", "The decomposition into strange scales is described by nested Xi sets without proving that an appropriate nested selection exists in these non-partition neighborhoods.", "The incidence and overlap issues from the random-part proof remain." ], "reasoning": "The terminal comparison using d_0 C_1=10 is consistent. It cannot certify the preceding lower and upper estimates." }, { "label": "Lemma 9.11", "element_type": "lemma", "element_id": "lemma_sigma_j", "source_label": "lem:sigma j lemma", "argument_gist": "Combines small relative contracting slope and late encounter geometry to localize an entire heavy slice in a smaller translated transverse neighborhood.", "dependencies": [ { "source_label": "lm:concentrated-surface", "evidence": "Then by Lemma~\\ref{lm:concentrated-surface}, we have $\\|\\vv w^+\\| \\le \\beta e^{s' - j}$." }, { "source_label": "lm:product-w", "evidence": "By Lemma \\ref{lm:product-w}," } ], "external_inputs": [], "external_references": [], "verdict": "major_issues", "issues": [ "The proof defines hat y_i using a(-2t), but all following estimates require a(-2s'). These conjugations have different transverse scales.", "The final relative block-diagonal radius is 2.1beta, whereas Pi_{s',j-1} still has radius beta in those directions. Decreasing j enlarges only the transverse radii and does not absorb this discrepancy.", "The contracting-coordinate estimates depend on the unresolved concentrated-surface lemma." ], "reasoning": "Replacing t by s' addresses one evident typographical error, but the remaining neighborhood containment still does not follow from the displayed bounds." }, { "label": "Proposition 10.1", "element_type": "proposition", "element_id": "proposition_dimension_no_drop", "source_label": "prop:dimension-does-not-drop", "argument_gist": "Repeats the normal and strange weighted-intersection estimates at a higher heaviness threshold to show that extremely heavy neighborhoods have small total mass.", "dependencies": [ { "source_label": "prop:dimension-improvement-random", "evidence": "which follows the proof of Proposition \\ref{prop:dimension-improvement-random}." }, { "source_label": "prop:dimension-improvement-strange-part", "evidence": "which follows the proof of Proposition \\ref{prop:dimension-improvement-strange-part}." }, { "source_label": "lm:concentrated-surface", "evidence": "For the upper bound, by Lemma \\ref{lm:concentrated-surface}, for any $y_1 \\in \\fJ \\cap \\Sigma^r$ and $z \\in \\cI \\cap \\Sigma^r \\cap \\fZ'$, we have that for any $y_2 \\in \\cI_z$ such that $(y_1, y_2) \\in \\diamond'_r$, we have" } ], "external_inputs": [], "external_references": [], "verdict": "major_issues", "issues": [ "This is only a sketch and inherits the unresolved geometry and conditioning estimates of both preceding incidence proofs.", "The definition of diamond'_{r,j} tests intersection with Z' rather than the strange-scale set Z'_j used to construct the lower bound.", "The count of heavy slices again requires bounded overlap rather than disjointness, and normalization factors are inconsistently omitted.", "Approximate comparison notation is used without quantified losses in estimates essential to the contradiction." ], "reasoning": "The higher-threshold strategy is intelligible and there is no apparent circular dependency: it is intended to precede the structured-part proof. Its proof is nevertheless incomplete." }, { "label": "Proposition 11.1", "element_type": "proposition", "element_id": "proposition_structured_dimension", "source_label": "prop:dimension-improvement-structured-component", "argument_gist": "Turns a large family of concentrated encounter curves into many occupied fine packets, unless the curves lie in narrow strips corresponding to SL(2,R) semidirect R^2 neighborhoods; rules out those strips by subgroup non-concentration.", "dependencies": [ { "source_label": "prop:dimension-does-not-drop", "evidence": "Let $\\fZ \\subset \\Pi(x)$ be the subset from Proposition \\ref{prop:dimension-does-not-drop}." }, { "source_label": "lm:integrable-surface", "evidence": "By Lemma \\ref{lm:integrable-surface}, we have that for every highly concentrated curve neighborhood $\\cC$, there exist at most" }, { "source_label": "lm:determine-curve-nbhd", "evidence": "By Lemma \\ref{lm:determine-curve-nbhd}, we have that the curve neighborhoods from $\\mathfrak{L}_n(\\cC, m_{\\cC})$ are contained" }, { "source_label": "lm:counting-curves-nbhd", "evidence": "By the last statement of Lemma \\ref{lm:counting-curves-nbhd}, we have that each curve neighborhood from $\\mathfrak{L}_n(\\cC, m_{\\cC})$ can cover at most" }, { "source_label": "lm:curve-in-N", "evidence": "Indeed, by Lemma \\ref{lm:curve-in-N}, there exists $z' \\in \\Theta'(z)$ so that the following holds:" }, { "source_label": "prop:sl2xr2-closing-lemma", "evidence": "By our discussion at the beginning of \\S \\ref{sec-reduction}, Proposition \\ref{prop:sl2xr2-closing-lemma} holds for $\\mu_{\\cF_i}$." } ], "external_inputs": [], "external_references": [], "verdict": "major_issues", "issues": [ "The curve-comparison package depends on the false implication in lm:hyperbola-plane and the unproved counting lemmas.", "The subgroup non-concentration input has a reversed conjugation in its proof.", "The asserted passage from a concentrated curve to at least e^{2s'-9C_1 epsilon_1 s'} full packets needs a separate estimate on almost-empty packets and overlap; it is not demonstrated.", "Curves passing within a neighborhood of a grid point are subsequently treated as exact hyperbolas with that common center. The parameter errors from this recentering are not controlled.", "Removing curves contained in one narrow strip does not imply that the next strip obtained is disjoint from the first. The repeated-strip construction and its count of distinct full packets require an additional selection argument.", "The greedy construction guarantees new points relative to earlier selected curves, not points uncovered by every other selected curve as later asserted. A disjoint charging scheme would have to be explicitly maintained." ], "reasoning": "This is the novel bottleneck of the proposed argument. The large exponent margins may accommodate some local losses, but they do not replace the missing geometric selection and counting arguments." }, { "label": "Lemma 11.4", "element_type": "lemma", "element_id": "lemma_hyperbola_plane", "source_label": "lm:hyperbola-plane", "argument_gist": "Relates closeness and intersection lengths of rational graphs X=-Y/(wY+k) to closeness of their parameters.", "dependencies": [], "external_inputs": [], "external_references": [], "verdict": "major_issues", "issues": [ "Part (1)'s necessary parameter bounds omit factors depending on beta and are false with the stated constants. Set ell=0, k=k'=1, w=beta e^{s'}/2 and w'=w+2e^{-s'}, with admissible sufficiently large s', and A=1. Across the box, graph displacement is O(beta^2 e^{-3s'}), hence less than e^{-3s'} for small beta, but |w-w'|=2e^{-s'} exceeds the asserted e^{-s'} bound.", "The proof treats information over a Y-interval of size beta e^{-s'} as though its size were e^{-s'}, precisely losing beta-dependent factors in the necessary estimates.", "Euclidean neighborhood inclusion is replaced by vertical graph comparison without proving a uniform equivalence.", "Several algebraic estimates divide by w-w' without separately treating equality.", "Part (2)'s reduction from a lower bound on the point norm to a lower bound on |Y| requires slope bounds for the second graph that are not established." ], "reasoning": "The counterexample uses graphs of uniformly bounded slope and denominators near 1, so the small graph-displacement bound also gives the required geometric closeness after clipping to the square. This disproves the literal quantitative implication, although a weakened estimate with beta losses could be investigated." }, { "label": "Lemma 11.5", "element_type": "lemma", "element_id": "lemma_integrable_surface", "source_label": "lm:integrable-surface", "argument_gist": "Counts packet labels inducing comparable encounter curves by fixing an approximate center, restricting rational-graph parameters, and counting admissible grid configurations.", "dependencies": [ { "source_label": "lm:hyperbola-plane", "evidence": "By Lemma \\ref{lm:hyperbola-plane}(1) with $A =2 e^{11C_1\\epsilon_1 s'}$, we have that" }, { "source_label": "rmk:concentrated-surface", "evidence": "In this case, by Remark \\ref{rmk:concentrated-surface}, we can consider the corresponding straight lines" } ], "external_inputs": [], "external_references": [], "verdict": "major_issues", "issues": [ "Uses the false necessary parameter estimate from lm:hyperbola-plane(1).", "The counts of y' in M(C,z_0) are asserted without deriving the relationship between the original packet grid and the parameters w,k,r.", "It is not proved that all comparable curves can be represented through one of the selected z_0 with the stipulated exact relation.", "There is an unexplained improvement from an 11C_1 epsilon_1 tolerance to a 10C_1 epsilon_1 tolerance in the proof." ], "reasoning": "The proposed parameter-counting mechanism is not a substitute for the omitted discretization estimates. The numerical bounds are unverified." }, { "label": "Lemma 11.6", "element_type": "lemma", "element_id": "lemma_determine_curve", "source_label": "lm:determine-curve-nbhd", "argument_gist": "Uses a noncentral intersection and a parameter-difference scale to place one encounter curve in a controlled neighborhood of another.", "dependencies": [ { "source_label": "lm:hyperbola-plane", "evidence": "By Lemma \\ref{lm:hyperbola-plane}(2) with $A = e^{2s' - 2\\ell -m}$ we have that $\\cC'$ is contained in the $e^{-s' - m - 2\\ell}$-neighborhood of $\\cC$." } ], "external_inputs": [], "external_references": [], "verdict": "major_issues", "issues": [ "The cited part (2) yields a beta^{-100}A e^{-3s'} neighborhood, not the stronger A e^{-3s'} neighborhood claimed here.", "A maximum bound on changes in v and w is replaced without proof by the same scale bound on w alone.", "The hypotheses needed to apply lm:hyperbola-plane to the primed parameters are not all stated.", "The proof depends on the unresolved graph-comparison lemma." ], "reasoning": "Even accepting the cited lemma provisionally, the displayed conclusion drops an explicit factor beta^{-100}. Thus the asserted inference is not valid as written." }, { "label": "Lemma 11.7", "element_type": "lemma", "element_id": "lemma_counting_curve_neighborhoods", "source_label": "lm:counting-curves-nbhd", "argument_gist": "Packs admissible rational-graph parameters at the comparability scale and uses a short-intersection estimate to bound occupied grid points.", "dependencies": [ { "source_label": "lm:hyperbola-plane", "evidence": "By Lemma \\ref{lm:hyperbola-plane} (1) with $A = e^{2s' - m - 2 \\ell}$, we have" } ], "external_inputs": [], "external_references": [], "verdict": "major_issues", "issues": [ "The parameter packing relies on the false necessity direction of lm:hyperbola-plane(1).", "The collection's stated hypotheses do not impose all the parameter-range assumptions later used for its members.", "The final number of grid points is inferred from an intersection neighborhood without a complete grid-density estimate, including the beta-dependent mesh and thickening.", "Replacing a maximum parameter difference by the w-difference needed in part (2) is again unproved." ], "reasoning": "A packing argument could in principle prove a related bound with appropriate parameter restrictions and losses. This proof does not establish the stated one." }, { "label": "Lemma 11.8", "element_type": "lemma", "element_id": "lemma_curve_in_subgroup_neighborhood", "source_label": "lm:curve-in-N", "argument_gist": "Converts confinement of an encounter curve to a narrow vertical strip into small fr_1 coordinates, identifying concentration near a translated SL(2,R) semidirect R^2 piece.", "dependencies": [ { "source_label": "lm:concentrated-surface", "evidence": "By Lemma \\ref{lm:concentrated-surface}, we have that $\\cC = \\cC(\\cT(y), z)$ is the $\\beta^{-100} e^{-3s'}$-neighborhood of the following curve ($v_1,w_1$ as above):" }, { "source_label": "lm:product-w", "evidence": "By Lemma \\ref{lm:product-w}, we have" } ], "external_inputs": [], "external_references": [], "verdict": "major_issues", "issues": [ "The asserted existence of Y with |Y-Y_1|>=beta e^{-s'} is not guaranteed on a centered interval of length beta e^{-s'}; a constant loss is necessary.", "The inference |k|>=e^ell from |w(Y-Y_1)+k|>=e^ell and |w(Y-Y_1)|<=beta^2 loses a subtraction.", "The BCH step is written for unscaled vectors but then uses bounds established only for their conjugated, beta-sized versions.", "The passage from one selected tilde y to the entire Omega'(y) needs a quantitative anisotropic product estimate, especially when ell approaches s'.", "The proof inherits the unproved encounter-curve representation." ], "reasoning": "The subgroup interpretation is a useful structural observation. The literal containment and constants do not follow from the supplied calculations." }, { "label": "Proposition 12.1", "element_type": "proposition", "element_id": "proposition_high_dimension_decomposition", "source_label": "prop:high-dim-decomposition", "argument_gist": "Conditions a high-dimensional orbit measure on anisotropic boxes, projects away contracting and block-diagonal coordinates, and seeks a transverse Frostman measure to which Proposition 3.1 applies.", "dependencies": [ { "source_label": "eq:fvy", "evidence": "Moreover, we may apply \\eqref{eq:fvy}, \\eqref{eq:fwy}, and \\eqref{eq:fy} to conclude that for" }, { "source_label": "eq:compare-u-orbit", "evidence": "By \\eqref{eq:compare-u-orbit}, we have that $a(2s')_\\ast\\mu_k$ and $a(2s')_{\\ast}\\bar \\mu_k$ are $O(\\beta)$-close, namely, for any $f \\in C_c^{\\infty} (X)$," } ], "external_inputs": [], "external_references": [], "verdict": "major_issues", "issues": [ "Equation eq:decompose-hat-mu is an exact positive-measure decomposition, but the proof replaces each component by a measure only O(beta)-close against C^1 tests. A weak signed error cannot be represented without proof as a small positive probability component.", "The Upsilon sets form a cover, not a partition. Summing their normalized restrictions with weights equal to their masses overcounts overlaps in eq:first-decompose-hat-mu.", "An arbitrary exponentially small deletion from an orbit need not leave a union of complete beta-length U-segments. The orbitwise construction and resulting transverse weights are not defined rigorously.", "The displayed formulas for integrating f against mu_k omit the point x_k, so f is incorrectly evaluated on a group element rather than a point of X.", "The pointwise comparison before eq:compare-u-orbit omits a(2s') on its right-hand orbit.", "Transporting the measure does not preserve the exact niceness bound without controlling preimages of enlarged neighborhoods.", "The base points must satisfy the thick-part hypothesis of Proposition 3.1; this is not stated in the proposition's conclusion." ], "reasoning": "The proof outlines a plausible weak approximation strategy, not the exact convex decomposition stated. A corrected theorem would need a precise test-function error, a genuine disintegration or partition, and quantitative control of the transported Frostman bounds." } ], "argument_gist": "The target is uniform effective equidistribution of a long E_23-unipotent segment unless a prescribed central diagonal translate has an exceptionally short vector or bivector. The proposed proof first obtains weak transverse dimension, then repeatedly increases it through a weighted Kakeya model. Random and intermediate-scale concentrated packets are treated by intersection counts; the structured case is meant to force concentration near an SL(2,R) semidirect R^2 orbit, excluded by the Diophantine alternative. A final projection to the expanding transverse plane and a van der Corput estimate should give equidistribution. The bottlenecks are the initial closing argument, the encounter-hyperbola geometry and weighted counting, and the final measure projection; none is fully established in the supplied text.", "argument_steps": [ { "title": "Equivalent formulations and applications", "gist": "Relate long U-segments to diagonal translates and reduce expanding lines and curves to that formulation. The equivalence is correct; the curve proof has an independent Taylor-scale error.", "claims": [ "thm:effective-U-orbit", "thm:main-thm", "cor:straight-line", "cor:curve" ] }, { "title": "Local Lie algebra calculus", "gist": "Decompose sl(3) into the block-diagonal algebra and two transverse modules, then establish local product-coordinate estimates.", "claims": [ "lm:lie-bracket", "lm:product-lie-algebra-1", "lm:product-w", "lm:two-theta", "lm:product-theta", "cor:product-theta", "lm:different-coordinate-1" ] }, { "title": "Non-divergence and initial dimension", "gist": "Discard bad thick-part excursions and derive transverse non-concentration by producing many arithmetic near-stabilizers. This phase imports quantitative non-divergence and an unspecified effective closing argument.", "claims": [ "prop:quantitative-nondivergence", "lm:separation-lemma-a", "prop:initial-bound-dimension", "prop:sl2xr2-closing-lemma" ] }, { "title": "Localization and Kakeya model", "gist": "Condition on playgrounds, cover them by coarse and fine anisotropic packets, and describe packet encounters through rational coordinate formulas and hyperbolas.", "claims": [ "prop:reduction", "lm:cross-section", "lm:initial-dimension-under-a-t", "lm:partition-1", "lm:partition-2", "lm:wv2r", "lm:omega-u", "lm:time-leave", "lm:concentrated-surface", "lm:omega-upper" ] }, { "title": "Unstructured incidence estimates", "gist": "Compare lower intersection counts forced by repeated visits to heavy packets with upper counts from the encounter curves. Separate normal packets from intermediate-scale strange packets.", "claims": [ "prop:dimension-improvement-random", "prop:dimension-improvement-strange-part", "lem:sigma j lemma", "prop:dimension-does-not-drop" ] }, { "title": "Structured incidence estimates", "gist": "Count comparable encounter curves and their intersections. If occupation fails to grow, use narrow strips to identify a forbidden subgroup neighborhood.", "claims": [ "lm:hyperbola-plane", "lm:integrable-surface", "lm:determine-curve-nbhd", "lm:counting-curves-nbhd", "lm:curve-in-N", "prop:dimension-improvement-structured-component", "prop:improve-dimension" ] }, { "title": "Iteration and conversion to equidistribution", "gist": "Iterate until transverse dimension is almost four, project to two expanding coordinates, and apply the externally motivated high-dimensional averaging estimate.", "claims": [ "prop:high-dim-decomposition", "prop:high-dimension-to-equidistribution-a", "thm:main-thm" ] } ], "external_references": [ { "bibliography_key": "ratner-acta", "role": "background", "locator": "", "audit_status": "not_inspected", "evidence": "The proof is inspired by Ratner's original proof of her measure rigidity theorem \\cite{ratner-acta} and recent papers by" }, { "bibliography_key": "ratner-invent", "role": "background", "locator": "", "audit_status": "not_inspected", "evidence": "(cf. \\cite{ratner-acta}, \\cite{ratner-invent}, \\cite{Ratner}) which proves a conjecture by Dani \\cite{Dani1981}," }, { "bibliography_key": "Ratner", "role": "background", "locator": "", "audit_status": "not_inspected", "evidence": "(cf. \\cite{ratner-acta}, \\cite{ratner-invent}, \\cite{Ratner}) which proves a conjecture by Dani \\cite{Dani1981}," }, { "bibliography_key": "Dani1981", "role": "background", "locator": "", "audit_status": "not_inspected", "evidence": "(cf. \\cite{ratner-acta}, \\cite{ratner-invent}, \\cite{Ratner}) which proves a conjecture by Dani \\cite{Dani1981}," }, { "bibliography_key": "lindenstrauss-margulis", "role": "background", "locator": "", "audit_status": "not_inspected", "evidence": "Lindenstrauss and Margulis proved \\cite{lindenstrauss-margulis} \n an effective density result for one-parameter unipotent orbits in $\\SL(3,\\R)/\\SL(3,\\Z)$ and applied it to prove an effective version of Oppenheim's conjecture." }, { "bibliography_key": "einsiedler-mohammadi", "role": "background", "locator": "", "audit_status": "not_inspected", "evidence": "see \\cite{lindenstrauss-margulis}, \\cite{einsiedler-mohammadi}, \\cite{LM-preprint2021}, \\cite{LMW-preprint2022}," }, { "bibliography_key": "LM-preprint2021", "role": "background", "locator": "", "audit_status": "not_inspected", "evidence": "Lindenstrauss, Mohammadi, and Wang \\cite{LM-preprint2021,LMW-preprint2022}." }, { "bibliography_key": "LMW-preprint2022", "role": "background", "locator": "", "audit_status": "not_inspected", "evidence": "Lindenstrauss, Mohammadi, and Wang \\cite{LM-preprint2021,LMW-preprint2022}." }, { "bibliography_key": "nelson-venkatesh", "role": "background", "locator": "", "audit_status": "not_inspected", "evidence": "\\cite{Venkatesh2010}, \\cite{nelson-venkatesh}, \\cite{einsiedler-margulis-venkatesh}, \\cite{chow-yang}, \\cite{Browning2016}" }, { "bibliography_key": "chow-yang", "role": "background", "locator": "", "audit_status": "not_inspected", "evidence": "Building on Strombergsson's result, Chow and the author \\cite{chow-yang} proved an" }, { "bibliography_key": "Browning2016", "role": "background", "locator": "", "audit_status": "not_inspected", "evidence": "The reader is also referred to \\cite{Browning2016} for a similar effective equidistibution result which has applications to number theory." }, { "bibliography_key": "green_tao2012", "role": "background", "locator": "", "audit_status": "not_inspected", "evidence": "For $G$ being nilpotent, the effective version of Ratner's theorem was established by Green and Tao \\cite{green_tao2012}." }, { "bibliography_key": "margulis-thesis", "role": "background", "locator": "", "audit_status": "not_inspected", "evidence": "one can apply the thickening argument developed in Margulis's thesis (cf. \\cite{margulis-thesis}, \\cite{Klein-Mar-Effective-Equid}) and" }, { "bibliography_key": "Klein-Mar-Effective-Equid", "role": "background", "locator": "", "audit_status": "not_inspected", "evidence": "See \\cite{Klein-Mar-Effective-Equid}, \\cite{edwards-effective} for results in this setting." }, { "bibliography_key": "sarnak1982", "role": "background", "locator": "", "audit_status": "not_inspected", "evidence": "See \\cite{sarnak1982}, \\cite{burger1990}, \\cite{Flaminio-Forni}, \\cite{Sarnak-Ubis}, \\cite{Strombergsson2004} and \\cite{Strombergsson2013} for works in this setting." }, { "bibliography_key": "burger1990", "role": "background", "locator": "", "audit_status": "not_inspected", "evidence": "See \\cite{sarnak1982}, \\cite{burger1990}, \\cite{Flaminio-Forni}, \\cite{Sarnak-Ubis}, \\cite{Strombergsson2004} and \\cite{Strombergsson2013} for works in this setting." }, { "bibliography_key": "Flaminio-Forni", "role": "background", "locator": "", "audit_status": "not_inspected", "evidence": "See \\cite{sarnak1982}, \\cite{burger1990}, \\cite{Flaminio-Forni}, \\cite{Sarnak-Ubis}, \\cite{Strombergsson2004} and \\cite{Strombergsson2013} for works in this setting." }, { "bibliography_key": "Sarnak-Ubis", "role": "background", "locator": "", "audit_status": "not_inspected", "evidence": "See \\cite{sarnak1982}, \\cite{burger1990}, \\cite{Flaminio-Forni}, \\cite{Sarnak-Ubis}, \\cite{Strombergsson2004} and \\cite{Strombergsson2013} for works in this setting." }, { "bibliography_key": "Strombergsson2004", "role": "background", "locator": "", "audit_status": "not_inspected", "evidence": "See \\cite{sarnak1982}, \\cite{burger1990}, \\cite{Flaminio-Forni}, \\cite{Sarnak-Ubis}, \\cite{Strombergsson2004} and \\cite{Strombergsson2013} for works in this setting." }, { "bibliography_key": "Strombergsson2013", "role": "background", "locator": "", "audit_status": "not_inspected", "evidence": "See \\cite{sarnak1982}, \\cite{burger1990}, \\cite{Flaminio-Forni}, \\cite{Sarnak-Ubis}, \\cite{Strombergsson2004} and \\cite{Strombergsson2013} for works in this setting." }, { "bibliography_key": "furstenberg1973", "role": "background", "locator": "", "audit_status": "not_inspected", "evidence": "In fact, equidistribution for this case was proved before Ratner's theorem, see \\cite{furstenberg1973}, \\cite{dani-smillie}." }, { "bibliography_key": "dani-smillie", "role": "background", "locator": "", "audit_status": "not_inspected", "evidence": "In fact, equidistribution for this case was proved before Ratner's theorem, see \\cite{furstenberg1973}, \\cite{dani-smillie}." }, { "bibliography_key": "edwards-effective", "role": "background", "locator": "", "audit_status": "not_inspected", "evidence": "See \\cite{Klein-Mar-Effective-Equid}, \\cite{edwards-effective} for results in this setting." }, { "bibliography_key": "Strombergsson2015", "role": "background", "locator": "", "audit_status": "not_inspected", "evidence": "Using techniques from analytic number theory, Strombergsson \\cite{Strombergsson2015}" }, { "bibliography_key": "kim2021", "role": "background", "locator": "", "audit_status": "not_inspected", "evidence": "Strombergsson's result was recently generalized by Kim \\cite{kim2021} to $\\SL(n, \\R) \\ltimes \\R^n/ \\SL(n,\\Z) \\ltimes \\Z^n$ with" }, { "bibliography_key": "einsiedler-margulis-venkatesh", "role": "background", "locator": "", "audit_status": "not_inspected", "evidence": "The reader is referred to \\cite{einsiedler-margulis-venkatesh} and \\cite{einsiedler-margulis-mohammadi-venkatesh}\n for effective equidistribution of closed orbits of maximal semisimple subgroups which can be regarded as an effective version" }, { "bibliography_key": "einsiedler-margulis-mohammadi-venkatesh", "role": "background", "locator": "", "audit_status": "not_inspected", "evidence": "The reader is referred to \\cite{einsiedler-margulis-venkatesh} and \\cite{einsiedler-margulis-mohammadi-venkatesh}\n for effective equidistribution of closed orbits of maximal semisimple subgroups which can be regarded as an effective version" }, { "bibliography_key": "Mozes_Shah", "role": "background", "locator": "", "audit_status": "not_inspected", "evidence": "of a result by Mozes and Shah \\cite{Mozes_Shah}." } ], "reference_role_notes": { "internally_proved_results": "The manuscript's own propositions remain internal dependencies, including Proposition 3.1; their citations to external arguments are recorded separately and do not convert the internal statements into independently verified imported theorems.", "explicitly_excluded_results": "No external result is explicitly declared unused in the supplied text. Contextual citations have not been relabelled as exclusions.", "retrospective_note": "No later theorem or later proof has been used to close an original proof obligation. The 2023\u20132024 bibliography entries remain background where used only historically; the actual proof use of LM Proposition 4.2 has unresolved pre-cutoff provenance.", "repair_status": "No replacement proof of a main claim is supplied in this review. Suggested sign, threshold, buffering and weak-error changes are possible repair directions, not completed repairs." }, "overall_assessment": { "validity_score": 35, "summary": "The manuscript presents a coherent and potentially valuable strategy, but the supplied text does not establish its main theorem. Several auxiliary statements or displayed inferences are demonstrably false, and the central incidence and measure-decomposition arguments contain substantial unverified steps; this assessment does not assert that the main theorem itself is false.", "critical_issues": [ "False non-divergence proposition with reversed endpoint hypothesis, and a separation lemma false in its stated parameter range.", "Unidentified quantitative closing input and unjustified arithmetic assertions in the initial dimension proof.", "Wrong-sign conjugation in the subgroup non-concentration proof.", "Impossible finite cover of the noncompact space X and incorrect handling of conditioned measures.", "Unproved packet geometry, overlap control and conditional orbit uniformity underlying the incidence estimates.", "False quantitative implication in lm:hyperbola-plane and unsupported consequences in the structured counting argument.", "An exact positive measure decomposition is inferred from weak approximation in Proposition 12.1.", "Order-one, rather than exponentially small, tangent approximation error in the curve corollary.", "Unresolved pre-cutoff version and adaptation of the external high-dimensional equidistribution argument." ], "minor_issues": [ "Norms, metrics, Sobolev order and the meaning of measure-valued O-errors should be fixed explicitly.", "Repeated normalization losses and enlarged neighborhood radii require systematic slack rather than exact inequalities.", "Several formulas omit base points or use inconsistent t and s' parameters.", "The displayed uniform Diophantine lower bound should be distinguished from the usual endpoint exponent terminology.", "A dated source version is needed to separate original submission content from later bibliographic updates." ], "strengths": [ "The long-orbit and expanding-segment formulations are correctly related.", "The Lie algebra decomposition and bracket structure provide a sound local framework.", "The random, strange and structured decomposition clearly identifies the intended mechanisms of dimension growth.", "The proposed encounter-curve model exposes a concrete geometric bottleneck rather than hiding the main difficulty behind qualitative rigidity.", "The final accumulated diagonal-time identity is consistent with the inductive schedule." ] }, "_meta": { "model": "gpt-6.1-sol", "effort": "high", "prompt_profile": "baseline-v3", "prompt_profile_sha256": "d5fb2af182eb3a766fef51c513c74eb0f8dfade127b3a6e7282b40bde9a1a337" }, "roadmap": { "schema_version": 1, "title": "Effective version of Ratner's equidistribution theorem for SL(3,R)", "source": { "name": "manuscript", "sha256": "e6f3ee5af078764b1750ea6a8cdbc8c1d9577ef6d14d0bc638830f385d839db8", "characters": 211793, "lines": 3562 }, "argument_gist": "The target is uniform effective equidistribution of a long E_23-unipotent segment unless a prescribed central diagonal translate has an exceptionally short vector or bivector. The proposed proof first obtains weak transverse dimension, then repeatedly increases it through a weighted Kakeya model. Random and intermediate-scale concentrated packets are treated by intersection counts; the structured case is meant to force concentration near an SL(2,R) semidirect R^2 orbit, excluded by the Diophantine alternative. A final projection to the expanding transverse plane and a van der Corput estimate should give equidistribution. The bottlenecks are the initial closing argument, the encounter-hyperbola geometry and weighted counting, and the final measure projection; none is fully established in the supplied text.", "review_provenance": { "pipeline": "full-text-batch", "model": "gpt-6.1-sol", "reasoning_effort": "high" }, "review_policy": { "schema": 1, "cutoff_date": "2022-09-26", "basis": "journal_received", "evidence": "Annals publisher page https://annals.math.princeton.edu/2025/202-1/p03, Milestones: Received 26 September 2022; cached publisher.html.", "status": "recorded", "rule": "external-results-available-by-cutoff" }, "phases": [ { "title": "Equivalent formulations and applications", "gist": "Relate long U-segments to diagonal translates and reduce expanding lines and curves to that formulation. The equivalence is correct; the curve proof has an independent Taylor-scale error.", "claims": [ "thm:effective-U-orbit", "thm:main-thm", "cor:straight-line", "cor:curve" ] }, { "title": "Local Lie algebra calculus", "gist": "Decompose sl(3) into the block-diagonal algebra and two transverse modules, then establish local product-coordinate estimates.", "claims": [ "lm:lie-bracket", "lm:product-lie-algebra-1", "lm:product-w", "lm:two-theta", "lm:product-theta", "cor:product-theta", "lm:different-coordinate-1" ] }, { "title": "Non-divergence and initial dimension", "gist": "Discard bad thick-part excursions and derive transverse non-concentration by producing many arithmetic near-stabilizers. This phase imports quantitative non-divergence and an unspecified effective closing argument.", "claims": [ "prop:quantitative-nondivergence", "lm:separation-lemma-a", "prop:initial-bound-dimension", "prop:sl2xr2-closing-lemma" ] }, { "title": "Localization and Kakeya model", "gist": "Condition on playgrounds, cover them by coarse and fine anisotropic packets, and describe packet encounters through rational coordinate formulas and hyperbolas.", "claims": [ "prop:reduction", "lm:cross-section", "lm:initial-dimension-under-a-t", "lm:partition-1", "lm:partition-2", "lm:wv2r", "lm:omega-u", "lm:time-leave", "lm:concentrated-surface", "lm:omega-upper" ] }, { "title": "Unstructured incidence estimates", "gist": "Compare lower intersection counts forced by repeated visits to heavy packets with upper counts from the encounter curves. Separate normal packets from intermediate-scale strange packets.", "claims": [ "prop:dimension-improvement-random", "prop:dimension-improvement-strange-part", "lem:sigma j lemma", "prop:dimension-does-not-drop" ] }, { "title": "Structured incidence estimates", "gist": "Count comparable encounter curves and their intersections. If occupation fails to grow, use narrow strips to identify a forbidden subgroup neighborhood.", "claims": [ "lm:hyperbola-plane", "lm:integrable-surface", "lm:determine-curve-nbhd", "lm:counting-curves-nbhd", "lm:curve-in-N", "prop:dimension-improvement-structured-component", "prop:improve-dimension" ] }, { "title": "Iteration and conversion to equidistribution", "gist": "Iterate until transverse dimension is almost four, project to two expanding coordinates, and apply the externally motivated high-dimensional averaging estimate.", "claims": [ "prop:high-dim-decomposition", "prop:high-dimension-to-equidistribution-a", "thm:main-thm" ] } ], "nodes": [ { "id": "sec-intro", "kind": "section", "label": "Section 1", "title": "Introduction", "section": "sec-intro", "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 377, "line_end": 623, "char_start": 9943, "char_end": 25217 } }, { "id": "thm:ratner", "kind": "thm", "label": "Thm 1.1", "title": "", "section": "sec-intro", "statement": "Let $G$ be a Lie group and $\\Gamma$ be a lattice in $G$, namely, the homogeneous space $X = G/\\Gamma$ admits a $G$-invariant probability measure. Let $U = \\{u(r): r\\in \\R\\}$ be a one-parameter unipotent subgroup of $G$. For any $x \\in X$, the closure of the $U$-orbit $U x$ of $x$ is a closed $L$-orbit $Lx$ for some Lie subgroup $L \\subset G$. Moreover, $Ux$ is equidistributed in $Lx$ with respect to the unique $L$-invariant probability measure $\\mu_L$, namely, for any $f \\in C^\\infty_c(X)$ (where $C^\\infty_c(X)$ denotes the space of smooth functions on $X$ with compact support), \\[ \\lim_{T \\to \\infty} \\frac{1}{T} \\int_{-T/2}^{T/2} f(u(r)x) \\dd r = \\int_{Lx} f \\dd \\mu_L .\\]", "argument_gist": "States the qualitative orbit-closure and equidistribution theorem to explain the effective problem. It is not used as a proved prerequisite of the effective argument.", "gist_origin": "review_model", "anchor": { "source": "manuscript", "line_start": 383, "line_end": 393, "char_start": 10184, "char_end": 10929 }, "review": { "verdict": "unverifiable", "issues": [ "No proof is supplied, and the external theorem was not inspected. This is a contextual statement, not an identified gap in the main proof." ], "reasoning": "The formulation is recognizable as the qualitative theorem, subject to the usual meaning of an Ad-unipotent flow for a general Lie group. Its inclusion does not certify the subsequent effective estimates." } }, { "id": "thm:effective-U-orbit", "kind": "thm", "label": "Thm 1.2", "title": "", "section": "sec-intro", "statement": "There exist $\\eta >0$ and $t_0 >1$ such that for any $t \\ge t_0$ and any $x_0 \\in X$, one of the following holds: \\par (1) For any $f \\in C^\\infty_c(X)$, we have \\[ \\left| \\int_{[1]} f(u(r e^t )x_0) \\dd r - \\int f \\dd \\mu_G \\right| \\ll e^{-\\eta t } \\|f\\|_S,\\] where $\\|\\cdot\\|_S$ denotes a fixed Sobolev norm; \\par (2) $a(-t) a_0( \\ell) x_0 \\not\\in X_{e^{-|\\ell|/3}}$ for some $ 0.99 t \\le |\\ell| \\le 1.01 t$.", "argument_gist": "Transfers the expanding-segment formulation to a long unipotent segment using a(t)u(r)=u(e^t r)a(t) and the commutation of a(t) with a_0(ell).", "gist_origin": "review_model", "anchor": { "source": "manuscript", "line_start": 480, "line_end": 488, "char_start": 17326, "char_end": 17801 }, "review": { "verdict": "major_issues", "issues": [ "The equivalence is correct, but the proof of thm:main-thm is not established; see its entry." ], "reasoning": "Substituting y_0=a(t)x_0 correctly transforms both alternatives. There is no additional independent proof of the effective assertion." } }, { "id": "thm:main-thm", "kind": "thm", "label": "Thm 1.3", "title": "", "section": "sec-intro", "statement": "Let $\\eta, t_0 >0$ be as above. For any $t \\ge t_0$, and any $x_0 \\in G/\\Gamma$, one of the following holds: \\par (1) For any $f \\in C^\\infty_c(X)$, we have \\[ \\left| \\int_{[1]} f(a(t) u(r) x_0 ) \\dd r - \\int f \\dd \\mu_G \\right| \\ll e^{-\\eta t} \\|f\\|_S;\\] \\par (2) $ a_0 (\\ell )x_0 \\not\\in X_{e^{-|\\ell|/3}}$ for some $ 0.99 t \\le |\\ell| \\le 1.01 t$.", "argument_gist": "Removes exceptional orbit portions, obtains initial transverse dimension, repeatedly improves that dimension through weighted Kakeya incidence estimates, and converts near-maximal transverse dimension into effective equidistribution.", "gist_origin": "review_model", "proof_anchor": { "source": "manuscript", "line_start": 3341, "line_end": 3353, "char_start": 202819, "char_end": 204209 }, "proof_opening": "\\par By Proposition \\ref{prop:initial-bound-dimension}, we have that $\\mu_{\\cF_0}$ is $(d_0, s_0)$-good. By Proposition \\ref{prop:dimension-improvement-random}, \\ref{prop:dimension-improvement-strange-part}, and \\ref{prop:dimension-improvement-structured-component} we get Proposition \\ref{prop:improve-dimension}. By Proposition \\ref{prop:reduction}, it implies that $\\mu_{\\cF_{i}}$ is $(d_i, s_i)$-good, where $d_i = d_0 + i \\epsilon_2$ and $s_i = 2^{-i}s_0$(we remove an exponentially small proportion at each step). We can proceed with this inductive construction until $d_i \\ge 4 - \\theta$. By Proposition \\ref{prop:high-dim-decomposition}, we have that \\[ a(2s_i)_\\ast \\mu_{\\cF_i} = \\sum_{k} \\n", "anchor": { "source": "manuscript", "line_start": 490, "line_end": 497, "char_start": 17900, "char_end": 18309 }, "review": { "verdict": "major_issues", "issues": [ "The foundational prop:quantitative-nondivergence is false as stated.", "The initial dimension and subgroup non-concentration arguments contain substantive gaps and, in prop:sl2xr2-closing-lemma, a reversed conjugation.", "The dimension-improvement chain uses incorrect conditioning inequalities and an unproved incidence geometry package; lm:hyperbola-plane contains a false quantitative implication.", "The final prop:high-dim-decomposition claims an exact positive measure decomposition but proves, at most, a weak approximation.", "The transfer of the external equidistribution estimate in prop:high-dimension-to-equidistribution-a remains unverified, including its pre-cutoff source version." ], "reasoning": "The terminal bookkeeping identity is consistent: the accumulated diagonal time before the final translate is t-2s_i. The problem is not that identity but the failure to establish several indispensable inputs. These objections do not constitute a counterexample to the main theorem itself." } }, { "id": "cor:straight-line", "kind": "cor", "label": "Cor 1.4", "title": "", "section": "sec-intro", "statement": "Let us denote $$a_1 (t) := a(t) a_0(t) = e^{t/3} & & \\\\ & e^{t/3} & \\\\ & & e^{-2t/3} .$$ Let $ \\vv w = (w_1, w_2) \\in \\R^2$ be a vector with Diophantine exponent $\\omega(\\vv w) = \\kappa \\le 0.6$, namely, for any positive integer $q \\in \\Z_+$, we have $$\\max \\{\\langle q w_1 \\rangle, \\langle q w_2 \\rangle\\} \\gg q^{-\\kappa},$$ where $\\langle \\cdot \\rangle$ denotes the distance to the nearest integer. For $\\vv v =(v_1, v_2) \\in \\R^2$ let us denote \\[ n(\\vv v) := 1 & & v_2 \\\\ & 1 & v_1 \\\\ & & 1 . \\] Let us define \\[\\varphi: [1] \\to \\R^2 \\] by $\\varphi(r) := (w_1 r + w_2 , r)$. Then there exist $ \\eta, t_0 >0$, such that for any $t \\ge t_0$ and any $f \\in C^\\infty_c(X)$ we have \\[ \\left| \\int_{[1]} f(a_1(t) n(\\varphi(r) )\\Gamma) \\dd r - \\int f \\dd \\mu_G \\right| \\ll e^{-\\eta t} \\|f\\|_S. \\]", "argument_gist": "Conjugates the expanding line into the main theorem's U-segment form and intends to rule out the exceptional alternative by the simultaneous Diophantine hypothesis on w.", "gist_origin": "review_model", "anchor": { "source": "manuscript", "line_start": 504, "line_end": 526, "char_start": 18642, "char_end": 19598 }, "review": { "verdict": "major_issues", "issues": [ "The main theorem is not established.", "The assertion that the Diophantine condition excludes every exceptional ell is not proved. Both the standard and exterior-square representations must be checked; the standard representation involves a dual linear-form estimate, not merely the displayed simultaneous lower bound.", "The stated definition uses a uniform lower bound at the endpoint exponent. This is stronger than merely specifying the usual supremal Diophantine exponent and should be distinguished terminologically." ], "reasoning": "The displayed matrix factorization and preservation of the integral under z(w_1) are correct. The remaining Diophantine assertion is plausible under the explicitly stated uniform hypothesis, but it is an omitted proof obligation rather than a verified step." } }, { "id": "rmk:expanding-line", "kind": "remark", "label": "Remark 1.5", "title": "", "section": "sec-intro", "statement": "The qualitative version of Corollary \\ref{cor:straight-line} was proved in \\cite{KNSY} for $\\omega(\\vv w) \\le 2$ using Ratner's theorem. The authors also proved that the equidistribution does not hold for $\\omega(\\vv w) > 2$.", "argument_gist": "The qualitative version of Corollary \\ref{cor:straight-line} was proved in \\cite{KNSY} for $\\omega(\\vv w) \\le 2$ using Ratner's theorem. The authors also proved that the equidistribution does not hold for $\\omega(\\vv w) > 2$.", "gist_origin": "extractive_statement", "anchor": { "source": "manuscript", "line_start": 548, "line_end": 557, "char_start": 20766, "char_end": 21259 } }, { "id": "cor:curve", "kind": "cor", "label": "Cor 1.6", "title": "", "section": "sec-intro", "statement": "Let us use the same notation as in Corollary \\ref{cor:straight-line}. Let $\\psi : [1] \\to \\R^2$ be a smooth non-degenerate curve in $\\R^2$, namely, derivatives of $\\psi$ span the whole space $\\R^2$ at every $r \\in [1]$. Then there exist $\\eta, t_0>0$ such that for any $t \\ge t_0$ and any $f \\in C^\\infty_c(X)$ we have \\[ \\left| \\int_{[1]} f(a_1(t) n(\\psi(r) )\\Gamma) \\dd r - \\int f \\dd \\mu_G \\right| \\ll e^{-\\eta t} \\|f\\|_S. \\]", "argument_gist": "Approximates short curve pieces by tangent U-segments, uses quantitative non-divergence to discard bad centers, and applies the main theorem to the remaining pieces.", "gist_origin": "review_model", "anchor": { "source": "manuscript", "line_start": 558, "line_end": 565, "char_start": 21261, "char_end": 21744 }, "review": { "verdict": "major_issues", "issues": [ "The claimed exponentially small tangent approximation at interval length e^{-t/2} is incorrect. For psi(r)=(r,r^2), writing r=r_0+e^{-t/2}v leaves a quadratic displacement v^2 in the E_13 coordinate after conjugation by a_1(t); it is of order one, not exponentially small.", "A general nondegenerate smooth curve is replaced by (r,psi_2(r)) without a local-chart decomposition, a treatment of the resulting density, or justification at points where the chosen coordinate derivative vanishes.", "The uniform exceptional-set estimate is asserted without checking the imported theorem's hypotheses.", "The bound for the union over all real ell does not follow from the bounds for individual ell. A discretization with an explicit compactness buffer is required.", "The proof changes [1] to [0,1] and does not justify how the good-center measure estimate yields a suitable partition into good short pieces." ], "reasoning": "The tangent factorization itself has the intended linear term. Its Taylor remainder has exactly the wrong size for the claimed approximation. This is a demonstrated error in the supplied corollary proof, independently of the issues with the main theorem." } }, { "id": "rmk:expanding-curve", "kind": "remark", "label": "Remark 1.7", "title": "", "section": "sec-intro", "statement": "The ineffective version of Corollary \\ref{cor:curve} was proved in \\cite{Shah_2} using Ratner's theorem.", "argument_gist": "The ineffective version of Corollary \\ref{cor:curve} was proved in \\cite{Shah_2} using Ratner's theorem.", "gist_origin": "extractive_statement", "anchor": { "source": "manuscript", "line_start": 599, "line_end": 603, "char_start": 23719, "char_end": 23883 } }, { "id": "sec-prelim", "kind": "section", "label": "Section 2", "title": "Preliminaries", "section": "sec-prelim", "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 623, "line_end": 885, "char_start": 25217, "char_end": 39992 } }, { "id": "lm:lie-bracket", "kind": "lemma", "label": "Lemma 2.1", "title": "", "section": "sec-prelim", "statement": "For $\\vv w^+ \\in \\vv r^+$, and $\\vv w^- \\in \\vv r^-$, \\[ [\\vv w^+, \\vv w^-] \\in \\R \\fa + \\R \\fa_0, \\] \\[ [\\vv w^+, \\fa] , [\\vv w^+, \\fa_0] \\in \\vv r^+, \\] \\[ [\\vv w^-, \\fa] , [\\vv w^-, \\fa_0] \\in \\vv r^-, \\] \\[ [\\vv w^-, \\fu] \\in \\vv r^+, [\\vv w^-, \\fu^\\ast] =\\vv 0, \\] and \\[ [\\vv w^+, \\fu^\\ast] \\in \\vv r^-, [\\vv w^+, \\fu] = \\vv 0. \\] For $\\vv w_1^+, \\vv w_2^+ \\in \\vv r^+$, \\[ [\\vv w_1^+, \\vv w_2^+] \\in \\R \\fu. \\] For $\\vv w_1^-, \\vv w_2^- \\in \\vv r^-$, \\[ [\\vv w_1^-, \\vv w_2^-] \\in \\R \\fu^\\ast. \\] For $\\vv h \\in \\fh$, \\[ [\\fa_0, \\vv h] = \\vv 0. \\] Also, we have \\[ [\\fa, \\fu] = \\fu, \\text{ } [\\fa, \\fu^\\ast] = - \\fu^\\ast, \\text{ and }[\\fu, \\fu^\\ast] = 2 \\fa. \\]", "argument_gist": "Computes brackets between the explicitly listed matrix units and diagonal generators.", "gist_origin": "review_model", "proof_anchor": { "source": "manuscript", "line_start": 751, "line_end": 754, "char_start": 29859, "char_end": 29949 }, "proof_opening": "\\par The statements can be verified by direct calculation.", "anchor": { "source": "manuscript", "line_start": 733, "line_end": 751, "char_start": 29071, "char_end": 29857 }, "review": { "verdict": "valid", "issues": [], "reasoning": "The inclusions and the three sl(2) identities follow directly from [E_ij,E_kl]=delta_jk E_il-delta_li E_kj. In particular, the transverse decomposition has the asserted bracket parity." } }, { "id": "lm:product-lie-algebra-1", "kind": "lemma", "label": "Lemma 2.2", "title": "", "section": "sec-prelim", "statement": "\\par Let $0 < \\rho_1, \\rho_2 < 10^{-10}$ be small constants. For $\\vv w \\in B_{\\fr_1 + \\fr_2}(\\rho_1)$ and $\\vv h \\in B_{\\R \\fa_0 + \\fh}(\\rho_2)$, we have that \\exp(\\vv w) \\exp(\\vv h) = \\exp(\\tilde{\\vv w} +\\tilde{\\vv h}), where $\\tilde{\\vv w} \\in \\fr_1 + \\fr_2$ with $\\|\\vv w - \\tilde{\\vv w}\\| \\le 2\\rho_2 \\|\\vv w\\|$, and $\\tilde{\\vv h} \\in \\R \\fa_0 + \\fh$ with $$\\|\\vv h - \\tilde{\\vv h} \\| \\le 2\\rho_1 \\|\\vv h\\|.$$ \\par Conversely, for $\\tilde{\\vv w} \\in B_{\\fr_1 + \\fr_2}(\\rho_1)$ and $\\tilde{\\vv h} \\in B_{\\R \\fa_0 + \\fh}(\\rho_2)$, we have \\exp(\\tilde{\\vv w} +\\tilde{\\vv h}) = \\exp(\\vv w) \\exp(\\vv h), where $\\vv w \\in \\fr_1 + \\fr_2$ with $\\|\\vv w - \\tilde{\\vv w}\\| \\le 4\\rho_2 \\|\\tilde {\\vv w}\\|$, and $\\vv h \\in \\R \\fa_0 + \\fh$ with $$\\|\\vv h - \\tilde{\\vv h} \\| \\le 4\\rho_1 \\|\\tilde{\\vv h}\\|.$$", "argument_gist": "Uses the local BCH series and the transverse/block-diagonal bracket decomposition, then inverts the local product coordinate map.", "gist_origin": "review_model", "proof_anchor": { "source": "manuscript", "line_start": 772, "line_end": 797, "char_start": 31127, "char_end": 33222 }, "proof_opening": "\\par Let $\\rho_0 = \\max\\left\\{\\rho_1, \\rho_2\\right\\}$. By the Baker-Campbell-Hausdorff formula, we have \\[ \\exp(\\vv w)\\exp(\\vv h) = \\exp\\left(\\vv w + \\vv h + P\\left(\\vv w, \\vv h\\right)\\right), \\] where P(\\vv w , \\vv h) =& \\frac{1}{2} \\left[ \\vv w, \\vv h \\right] + \\frac{1}{12} \\left( \\left[ \\vv w, \\left[\\vv w, \\vv h\\right] \\right] + \\left[ \\vv h, \\left[\\vv h, \\vv w\\right] \\right]\\right) + \\cdots. By Lemma \\ref{lm:lie-bracket} and the fact that for any $\\vv w_1, \\vv w_2 \\in \\fg$, $\\|[\\vv w_1, \\vv w_2]\\| \\le \\|\\vv w_1\\| \\cdot \\|\\vv w_2\\|$, we have that for any monomial $[\\cdots \\diamond \\cdots]$ in $P(\\vv w, \\vv h)$ with $i$ Lie brackets, we have $[\\cdots \\diamond \\cdots] \\in \\fr_1 + \\fr_2$ or ", "anchor": { "source": "manuscript", "line_start": 755, "line_end": 772, "char_start": 30069, "char_end": 31125 }, "review": { "verdict": "minor_issues", "issues": [ "A Lie algebra norm with the claimed bracket bound has not been specified.", "The local product-coordinate existence in the converse should be justified by the inverse function theorem.", "The last constant 4 does not follow from merely substituting the displayed coarse bounds ||w||<=2||tilde w|| and ||h||<=2||tilde h|| into the preceding estimate; tighter small-neighborhood bounds are needed." ], "reasoning": "The local statement is supported by the BCH expansion and the invertibility of the derivative of the product map. The quantitative constants need a fixed norm and slightly more careful bookkeeping; no structural obstruction was found." } }, { "id": "lm:product-w", "kind": "lemma", "label": "Lemma 2.3", "title": "", "section": "sec-prelim", "statement": "Let $0 < \\rho < \\rho_0 < 10^{-10}$ be two small constants. Let $\\vv w, \\vv w' \\in B_{\\fr_1 + \\fr_2} (\\rho_0)$ be such that \\[ \\|[\\vv w, \\vv w']\\| \\le \\rho. \\] Then \\[ \\exp(\\vv w ) \\exp( \\vv w') = \\exp(\\tilde{\\vv w}) \\exp( \\vv h), \\] where $\\tilde{\\vv w} \\in \\fr_1 + \\fr_2$ with $\\| \\tilde{\\vv w} - \\vv w - \\vv w'\\| \\le 26\\rho \\rho_0$, and $\\vv h \\in B_{\\R \\fa_0 + \\fh}(4\\rho)$.", "argument_gist": "Separates the even and odd transverse-degree terms in BCH and uses the preceding product-coordinate lemma.", "gist_origin": "review_model", "proof_anchor": { "source": "manuscript", "line_start": 805, "line_end": 844, "char_start": 33659, "char_end": 36133 }, "proof_opening": "\\par By the Baker-Campbell-Hausdorff formula, we have \\[ \\exp(\\vv w)\\exp(\\vv w') = \\exp\\left(\\vv w + \\vv w' + P\\left(\\vv w, \\vv w'\\right)\\right), \\] where P(\\vv w , \\vv w') =& \\frac{1}{2} \\left[ \\vv w, \\vv w' \\right] + \\frac{1}{12} \\left( \\left[ \\vv w, \\left[\\vv w, \\vv w'\\right] \\right] + \\left[ \\vv w', \\left[\\vv w', \\vv w\\right] \\right]\\right) + \\cdots. \\par By Lemma \\ref{lm:lie-bracket}, we have the following observations: \\par For $i = 2j+1$, every monomial $[\\cdots\\diamond\\cdots]$ in $P(\\vv w, \\vv w')$ with $i$ Lie brackets is in $\\R \\fa_0 + \\fh$. For $i = 2j$, every monomial $[\\cdots \\diamond \\cdots]$ in $P(\\vv w, \\vv w')$ with $i$ Lie brackets is in $\\fr_1 + \\fr_2$. \\par Moreover, by t", "anchor": { "source": "manuscript", "line_start": 797, "line_end": 805, "char_start": 33224, "char_end": 33657 }, "review": { "verdict": "minor_issues", "issues": [ "The BCH tail bounds and numerical constants depend on an unstated norm convention.", "The application of lm:product-lie-algebra-1 uses a radius up to 3rho_0, although that lemma is stated only for radii below 10^{-10}; the smallness threshold should be adjusted." ], "reasoning": "The bracket parity is correct, and the orders O(rho rho_0) and O(rho) follow locally. The issues concern the literal thresholds and constants." } }, { "id": "sec-high-dim-to-equidistribution", "kind": "section", "label": "Section 3", "title": "High transversal dimension to effective equidistribution", "section": "sec-high-dim-to-equidistribution", "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 885, "line_end": 915, "char_start": 39992, "char_end": 41897 } }, { "id": "prop:high-dimension-to-equidistribution-a", "kind": "prop", "label": "Prop 3.1", "title": "", "section": "sec-high-dim-to-equidistribution", "statement": "There exists $ \\theta >0$ such that the following holds: For any $0 < \\epsilon < 10^{-1000}$, there exist $ \\eta >0$ and $s_0 \\gg 1$ depending on $\\epsilon >0$ such that for any $s' \\ge s_0$, with $ e^{-\\epsilon^2 s'} \\le \\beta \\le 10^{-10}$, any Borel probability measure $\\rho$ on $[\\beta]^2$ with dimension larger than $2-2\\theta$ at scale $s'$, that is, for any square $S \\subset [\\beta]^2$ of size $\\beta e^{-s'}$, we have \\[ \\rho(S) \\le \\left(\\beta e^{-s'}\\right)^{2-2\\theta},\\] any function $f \\in C_c^\\infty(X)$, and any $x \\in X_{\\beta^{1/4}}$, we have \\[ \\left| \\int_{[\\beta]^2} \\int_{[1]} f(a(2s')\\exp( v_1 \\vv v_1 + w_1 \\vv w_1)u(\\beta r) x) \\dd r \\dd \\rho(w_1, v_1) - \\int f \\dd \\mu_G \\right| \\ll e^{-\\eta s'} \\|f\\|_S.\\]", "argument_gist": "Proposes a van der Corput and mixing argument converting a nearly two-dimensional distribution in the expanding transverse plane into equidistribution.", "gist_origin": "review_model", "proof_anchor": { "source": "manuscript", "line_start": 902, "line_end": 905, "char_start": 41182, "char_end": 41322 }, "proof_opening": "\\par The statement can be proved by following the proof of \\cite[Proposition 4.2]{LM-preprint2021} step by step.", "anchor": { "source": "manuscript", "line_start": 891, "line_end": 901, "char_start": 40354, "char_end": 41180 }, "review": { "verdict": "unverifiable", "issues": [ "There is no supplied proof beyond a reference to another proof in a different homogeneous-space setting.", "The necessary adaptation, constants, Sobolev norm and quantitative mixing input are not identified.", "Availability of the needed LM Proposition 4.2 before 2022-09-26 is unresolved." ], "reasoning": "This is a central proof obligation, not a routine citation that can be certified without inspecting and adapting the source. No counterexample to the proposition is established here." } }, { "id": "sec-quantitative-nondivergence", "kind": "section", "label": "Section 4", "title": "Quantitative non-divergence", "section": "sec-quantitative-nondivergence", "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 915, "line_end": 975, "char_start": 41897, "char_end": 45850 } }, { "id": "prop:quantitative-nondivergence", "kind": "prop", "label": "Prop 4.1", "title": "", "section": "sec-quantitative-nondivergence", "statement": "There exists $ \\alpha > 0$ such that the following holds: For any $\\beta, s >0$, if $x \\in X$ satisfies that $ a_0(-s)x , a_0(s)x \\not\\in X_{e^{- s/3 }}$, then, \\[ \\Leb\\left(\\left\\{r \\in [1]: a(s)u(r )x \\not\\in X_\\beta \\right\\}\\right) \\ll \\beta^{\\alpha}. \\]", "argument_gist": "Intends to use quantitative non-divergence to reduce cusp escape to a uniformly short vector or bivector, then detect that obstruction at an a_0 endpoint.", "gist_origin": "review_model", "proof_anchor": { "source": "manuscript", "line_start": 925, "line_end": 942, "char_start": 42495, "char_end": 43967 }, "proof_opening": "\\par Let us denote $x = g \\Gamma$. By the Kleinbock-Margulis quantitative non-divergence theorem (cf. \\cite[Theorem 2.2]{kleinbock2008}, \\cite[Theorem 5.2]{Klein_Mar}), if the statement does not hold then there exists a nonzero $\\vv v \\in \\bigwedge^j \\Z^3$ (where $j = 1$ or $2$), such that \\[ \\max_{r \\in [1]}\\| a(s) u(r) g \\vv v\\| \\le 1. \\] \\par Case 1. $\\vv v \\in \\Z^3$: Let us denote $ \\vv v' = a(s) g \\vv v = (v'_1, v'_2, v'_3)$. Then \\[ a(s) u(r) g \\vv v = u(r e^{s}) \\vv v' = (v'_1 , v'_2 + r e^{s} v'_3, v'_3). \\] Then $\\|a(s) u(r ) g \\vv v\\| \\le 1$ implies that $|v'_1| \\le 1 $, $|v'_2 + r e^{s} v'_3| \\le 1$ for any $r \\in [1]$. The latter easily implies that $|v'_2| \\le 1$ and $|v'_3| \\le", "anchor": { "source": "manuscript", "line_start": 918, "line_end": 925, "char_start": 42160, "char_end": 42494 }, "review": { "verdict": "major_issues", "issues": [ "The proposition is false: its hypothesis says that both a_0 endpoints are outside the compact set, whereas the intended contradiction requires endpoint non-escape.", "Explicit counterexample: fix s=1, let g=diag(e^{-M},e^{M/2},e^{M/2}), and x=gGamma. For M>2/3 both endpoint lattices have e_1 shorter than e^{-1/3}. For any beta>e^{-M}, every a(1)u(r)x is outside X_beta, so the bad set has measure 1. Letting beta tend to zero and choosing M accordingly contradicts any uniform C beta^alpha bound.", "The proof's estimates also lose constants: on [-1/2,1/2], a linear expression bounded by 1 yields |v'_3|<=2e^{-s}, not the asserted e^{-s}. The final norm estimate does not follow from componentwise bounds with constant 1." ], "reasoning": "The counterexample uses a vector fixed by U and a(s), so it is independent of any uncertainty about the external theorem. Replacing the endpoint signs is an intended correction, not a verified repair of all quantitative details." } }, { "id": "lm:separation-lemma-a", "kind": "lemma", "label": "Lemma 4.2", "title": "", "section": "sec-quantitative-nondivergence", "statement": "For any $\\beta >0$, any $\\ell \\ge 1$, and any $x \\in X$ with $a(-\\ell)x \\in X_{\\beta^{1/4}}$, if \\[ \\exp( r_1 e^\\ell \\fu + r_2 e^{\\ell/2} \\vv v_1 + r_3 e^{\\ell/2} \\vv w_1) x = \\exp(\\vv v)x \\] where $ r_1 , r_2, r_3 \\in [\\beta]$, $\\|(r_1, r_2, r_3)\\| \\ge \\beta/4$, and $\\vv v \\in B_\\fg (\\beta)$. Let us write \\[ \\vv v = u \\fu + u^\\ast \\fu^\\ast + a \\fa + a_0 \\fa_0 + \\sum_{j=1}^2 w_j \\vv w_j + v_j \\vv v_j. \\] if we have $|u^\\ast| < \\beta e^{-\\ell}$, then $\\|(v_2, w_2)\\| \\ge \\beta e^{-\\ell/2}$.", "argument_gist": "Conjugates an apparent near-return by a(-ell) and intends to contradict the injectivity radius in the thick part.", "gist_origin": "review_model", "proof_anchor": { "source": "manuscript", "line_start": 958, "line_end": 974, "char_start": 44729, "char_end": 45848 }, "proof_opening": "\\par For a contradiction, let us assume that $|u^\\ast|<\\beta e^{-\\ell}, |v_2, w_2| < \\beta e^{-\\ell/2}$. Let us denote $x = g\\Gamma$. Then \\[ \\exp( r_1 e^\\ell \\fu + r_2 e^{\\ell/2} \\vv v_1 + r_3 e^{\\ell/2} \\vv w_1) g = \\exp(\\vv v)g \\gamma, \\] for some $\\gamma \\in \\Gamma$. For $\\ell >1$ large enough, we have $\\gamma \\neq e$. Then we have \\[ \\exp(-\\vv v) \\exp( r_1 e^\\ell \\fu + r_2 e^{\\ell/2} \\vv v_1 + r_3 e^{\\ell/2} \\vv w_1) = g \\gamma g^{-1}. \\] Then a(-\\ell) g \\gamma g^{-1} a(\\ell) &= a(-\\ell)\\exp(-\\vv v) \\exp( r_1 e^\\ell \\fu + r_2 e^\\ell \\vv w_1) a(\\ell) \\\\ &= \\exp(- \\Ad (a(-\\ell))\\vv v ) \\exp (r_1 \\fu + r_2 \\vv v_1 + r_3 \\vv w_1). Since $|u^\\ast| < \\beta e^{-\\ell}, |v_2, w_2| < \\beta e^{-\\e", "anchor": { "source": "manuscript", "line_start": 949, "line_end": 958, "char_start": 44161, "char_end": 44728 }, "review": { "verdict": "major_issues", "issues": [ "The lemma is false for the full stated range ell>=1. With the canonical coordinate norm, take ell=1, x=Gamma, r_1=beta/4, r_2=r_3=0 and v=(e beta/4)fu. For sufficiently small beta, a(-1)Gamma lies in X_{beta^{1/4}}, all hypotheses hold, and u*=v_2=w_2=0 violates the conclusion.", "The proof itself invokes ell sufficiently large to ensure gamma is nontrivial, a condition absent from the statement.", "The asserted membership in B^G(beta) requires constant control on a product of two beta-sized factors; it does not follow literally from the displayed estimates." ], "reasoning": "The counterexample is an equality already in the group, with gamma equal to the identity. A sufficiently-large-ell formulation may avoid this example, but that would be a changed statement." } }, { "id": "subsec-constants", "kind": "section", "label": "Section 5", "title": "List of Constants", "section": "subsec-constants", "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 975, "line_end": 993, "char_start": 45850, "char_end": 47153 } }, { "id": "sec-initial-dim-control", "kind": "section", "label": "Section 6", "title": "Initial Dimension Control", "section": "sec-initial-dim-control", "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 993, "line_end": 1197, "char_start": 47153, "char_end": 60719 } }, { "id": "def:initial-construction", "kind": "defn", "label": "Defn 6.1", "title": "", "section": "sec-initial-dim-control", "statement": "\\par $\\cF \\subset \\cE$ is defined by removing from $\\cE$ points $y_0 = a ( s) u(r)x_{0} \\in \\cE$ satisfying $ a_0(\\ell') a(-\\ell) y_0 \\not\\in X_{\\beta^{1/4}}$ for some $0 \\le \\ell \\le 2\\delta_2 t$ and $ - 2\\delta_1 t \\le \\ell' \\le 2\\delta_1 t$. Let $\\mu_{\\cF}$ denote the normalized $U$-orbit measure on $\\cF$.", "argument_gist": "\\par $\\cF \\subset \\cE$ is defined by removing from $\\cE$ points $y_0 = a ( s) u(r)x_{0} \\in \\cE$ satisfying $ a_0(\\ell') a(-\\ell) y_0 \\not\\in X_{\\beta^{1/4}}$ for some $0 \\le \\ell \\le 2\\delta_2 t$ and $ - 2\\delta_1 t \\le \\ell' \\le 2\\delta_1 t$. Let $\\mu_{\\cF}$ denote the normalized $U$-orbit measure on $\\cF$.", "gist_origin": "extractive_statement", "anchor": { "source": "manuscript", "line_start": 1032, "line_end": 1039, "char_start": 49067, "char_end": 49437 } }, { "id": "prop:initial-bound-dimension", "kind": "prop", "label": "Prop 6.2", "title": "", "section": "sec-initial-dim-control", "statement": "For any $\\ell \\in [\\delta_3 t, \\delta_1 t]$, and any $x \\in \\cF$, we have \\[ \\mu_{\\cF} (Q_\\ell(\\beta) B^{A_0 H}(\\beta) x) \\le e^{- d_0 \\ell}. \\]", "argument_gist": "Turns excessive local concentration into many short near-returns, extracts integer matrices almost stabilizing the H-volume vector, and seeks a nearby exact stabilizer producing a short lattice vector in the lower plane.", "gist_origin": "review_model", "proof_anchor": { "source": "manuscript", "line_start": 1061, "line_end": 1167, "char_start": 50162, "char_end": 58853 }, "proof_opening": "\\par Let $\\eta_1 = 100 d_0$. Let us divide $a(s)u([1])x_0$ into pieces of length $e^{\\eta_1 \\ell}$. \\par It suffices to show that for each piece $\\tilde \\cE$ from the above partition with $\\tilde \\cF := \\tilde \\cE \\cap \\cF \\neq \\emptyset$, and any $x \\in \\tilde \\cF$, \\[ |\\tilde \\cF \\cap Q_\\ell(\\beta) B^{A_0 H}(\\beta) x| \\le e^{- d_0 \\ell} e^{\\eta_1 \\ell}. \\] \\par For a contradiction, let us assume that the statement does not hold for some $x\\in \\tilde \\cF$. Then we can find at least $e^{\\eta_2 \\ell}$ (where $\\eta_2 = \\eta_1/4$) many $\\bar x \\in \\tilde \\cF \\cap Q_\\ell(\\beta) B^{A_0 H}(\\beta) x$, $r_{\\bar x} \\in [1]$ such that \\[ u(r_{\\bar x} e^{2 d_0 \\ell}) \\bar x = \\exp (\\vv v_{\\bar x}) a_0^", "anchor": { "source": "manuscript", "line_start": 1053, "line_end": 1058, "char_start": 49944, "char_end": 50158 }, "review": { "verdict": "major_issues", "issues": [ "The construction of F and its asserted small discarded mass depend on the false prop:quantitative-nondivergence.", "The packing argument producing the stated number of separated near-returns, including a lower bound on their return times, is not supplied.", "Equation eq:a0ge bounds the coefficient a_0 of fa, but the subsequent distinctness argument invokes it as a lower bound on a_2, the fu* coefficient. The cited separation lemma would concern the latter coefficient, not the former.", "The assertion that an element of SL(3,Z) with nontrivial unipotent part must itself be unipotent is false. The matrix with rows (-1,1,0), (0,-1,0), (0,0,1) is a counterexample.", "The claimed stabilizer Stab(v_H)=A_0H omits a disconnected component: diag(-1,-1,1) fixes the H-volume vector but is not in A_0H.", "The effective closing estimates, especially simultaneous closing in Case 2, are not justified by an identified external result.", "Bounds for a bounded representative g and losses under Ad(g^{-1}) are used without being established at the point where g is chosen." ], "reasoning": "Some local mistakes could be corrected separately, for example by changing a coefficient or taking a power of an integer matrix. Such changes do not supply the missing packing and effective closing arguments. The proposition is therefore unproved here, not disproved by the auxiliary matrix examples." } }, { "id": "prop:sl2xr2-closing-lemma", "kind": "prop", "label": "Prop 6.3", "title": "", "section": "sec-initial-dim-control", "statement": "For any $x \\in \\cF$, $j = 1,2$, and any $\\ell \\in [2\\delta_3 t, \\delta_1 t]$ \\[ \\mu_{\\cF}(Q(\\fr_j, \\ell, \\beta)B^{A_0 H}(\\beta)x) \\le e^{- d_0 \\ell/2}.\\]", "argument_gist": "Attempts to turn concentration near one transverse summand into concentration near both summands by applying a central diagonal translate, then repeats the initial closing argument.", "gist_origin": "review_model", "proof_anchor": { "source": "manuscript", "line_start": 1175, "line_end": 1190, "char_start": 59221, "char_end": 60146 }, "proof_opening": "\\par Let us prove the statement for $j = 1$. The proof for $j=2$ is similar. \\par For a contradiction, let us assume that there exists $x \\in \\cF$ such that \\[ \\mu_{\\cF}\\left(Q(\\fr_1, \\ell, \\beta)B^{A_0 H}(\\beta)x\\right) > e^{- d_0 \\ell/2}. \\] Let us denote $\\tilde x = a_0(\\ell) x$ and $\\hat \\mu = a_0(\\ell)_\\ast \\mu_{\\cF}$. Noting that $$a_0(\\ell) Q(\\fr_1, \\ell, \\beta)B^{A_0 H}(\\beta)x = Q_{\\ell/2}(\\beta) B^{A_0 H}(\\beta)\\tilde x,$$ we have that \\[ \\hat \\mu \\left(Q_{\\tilde \\ell}(\\beta) B^{A_0 H}(\\beta)\\tilde x\\right) > e^{- d_0 \\tilde \\ell}, \\] where $\\tilde \\ell = \\ell/2 \\in [\\delta_3 t, \\delta_1 t]$. By repeating the argument in the proof of Proposition \\ref{prop:initial-bound-dimension}, ", "anchor": { "source": "manuscript", "line_start": 1170, "line_end": 1175, "char_start": 59001, "char_end": 59219 }, "review": { "verdict": "major_issues", "issues": [ "The displayed conjugation identity has the wrong sign. On fr_1, Ad(a_0(ell)) contracts by e^{-ell/2}; on fr_2 it expands by e^{ell/2}. Thus Q(fr_1,ell,beta) is sent to transverse sizes beta e^{-3ell/2} and beta e^{ell/2}, not beta e^{-ell/2} in both summands.", "The proposed repeated proof inherits the unresolved initial closing argument and exceptional-set construction.", "The proof must also verify all thick-part conditions for the translated measure, not only its center." ], "reasoning": "For j=1 the equalizing conjugation would have the opposite sign. This is a demonstrated error in the supplied reduction; a corrected conjugation would still require a complete proof of the preceding proposition." } }, { "id": "sec-reduction", "kind": "section", "label": "Section 7", "title": "Basic Reduction", "section": "sec-reduction", "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 1197, "line_end": 1285, "char_start": 60719, "char_end": 66700 } }, { "id": "def:dimension-control", "kind": "defn", "label": "Defn 7.1", "title": "", "section": "sec-reduction", "statement": "Given a probability measure $\\mu$ on $X$, we say that $\\mu$ is $(d, \\ell)$-good if for any $x_0 \\in X$, \\[\\mu (Q_\\ell(\\beta) B^{A_0 H}(\\beta) x_0) \\le e^{- d \\ell}.\\] We say that $\\mu$ is $(d, [\\ell_1, \\ell_2])$-good if it is $(d, \\ell)$-good for any $\\ell \\in [\\ell_1 , \\ell_2]$.", "argument_gist": "Given a probability measure $\\mu$ on $X$, we say that $\\mu$ is $(d, \\ell)$-good if for any $x_0 \\in X$, \\[\\mu (Q_\\ell(\\beta) B^{A_0 H}(\\beta) x_0) \\le e^{- d \\ell}.\\] We say that $\\mu$ is $(d, [\\ell_1, \\ell_2])$-good if it is $(d, \\ell)$-good for any $\\ell \\in [\\ell_1 , \\ell_2]$.", "gist_origin": "extractive_statement", "anchor": { "source": "manuscript", "line_start": 1202, "line_end": 1209, "char_start": 61204, "char_end": 61549 } }, { "id": "prop:reduction", "kind": "prop", "label": "Prop 7.2", "title": "", "section": "sec-reduction", "statement": "For $x \\in X$, let us call a neighborhood of the form $\\Pi(x) = \\Pi_{s'} x$ a playground. A playground $\\Pi(x)$ is called a full playground if \\[ \\mu(\\Pi(x)) \\ge \\beta^{15}. \\] For any full playground $\\Pi(x)$, let us denote \\[ \\mu_{\\Pi(x)} := \\mu(\\Pi(x))^{-1} \\mu|_{\\Pi(x)}. \\] If for any full playground $\\Pi(x)$, there exists \\[ \\cR_x \\subset \\Pi(x) \\] such that $$\\mu_{\\Pi(x)} (\\cR_x) \\le \\beta^{20},$$ and after removing $\\cR_x$, $\\mu_{\\Pi(x)}$ is $(d_i+2\\epsilon_2, s_{i+1})$-good. Then there exists $\\cR_{\\diamond} \\subset X$ such that \\[ \\mu(\\cR_{\\diamond}) \\le \\beta^2, \\] and after removing $\\cR_{\\diamond}$, $\\mu$ is $(d_i + \\epsilon_2, s_{i+1})$-good.", "argument_gist": "Covers the measure by full playgrounds, discards light playgrounds and local exceptional sets, and absorbs a covering multiplicity into the dimension-improvement margin.", "gist_origin": "review_model", "proof_anchor": { "source": "manuscript", "line_start": 1251, "line_end": 1276, "char_start": 64097, "char_end": 65711 }, "proof_opening": "\\par Note that there exists a finite subset $\\fQ \\subset X$ with $|\\fQ| \\le \\beta^{-9}$ and \\[ X = \\bigcup_{x \\in \\fQ} \\Pi(x). \\] Let $\\fQ_1 \\subset \\fQ$ denote the set of $x \\in \\fQ$ such that $\\Pi(x)$ is a full playground, and $\\fQ_2 = \\fQ \\setminus \\fQ_1$. Then \\[ \\mu \\left( \\bigcup_{x \\in \\fQ_2} \\Pi(x) \\right) \\le \\beta^{-9} \\times \\beta^{15} \\le \\beta^{6}. \\] By the given condition, for every $x \\in \\fQ_1$, we get $\\cR_x \\subset \\Pi(x)$ with \\[ \\mu_{\\Pi(x)} (\\cR_x) \\le \\beta^{20}, \\] and after removing $\\cR_x$, $\\mu_{\\Pi(x)}$ is $(d_i + 2 \\epsilon_2, s_{i+1})$-good. Then \\[ \\mu\\left( \\bigcup_{x \\in \\fQ_1} \\cR_x\\right) \\le \\beta^{-9} \\times \\beta^{20} = \\beta^{11}. \\] Let \\[ \\cR_{\\diamon", "anchor": { "source": "manuscript", "line_start": 1237, "line_end": 1251, "char_start": 63380, "char_end": 64096 }, "review": { "verdict": "major_issues", "issues": [ "The first asserted finite cover of all X by at most beta^{-9} translates of Pi_{s'} is impossible. Pi_{s'} is compact for fixed parameters, so a finite union of its translates cannot cover the noncompact space X.", "A cover of an appropriate thick support might replace this claim, but its cardinality and the choice of centers must be proved.", "The proof switches between restricted measures and renormalized probability measures. Removing further mass does not preserve an exact probability upper bound without a normalization loss." ], "reasoning": "The union-bound idea is sound on a suitably controlled compact support with slack. The actual cover asserted in the proof is false and the required replacement is not supplied." } }, { "id": "sec-kakeya-model", "kind": "section", "label": "Section 8", "title": "Construction of a Kakeya-type model", "section": "sec-kakeya-model", "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 1285, "line_end": 1958, "char_start": 66700, "char_end": 109617 } }, { "id": "lm:two-theta", "kind": "lemma", "label": "Lemma 8.1", "title": "", "section": "sec-kakeya-model", "statement": "\\par Let us denote \\[ \\hat{\\Theta}_{s'} := Q_{s'}(1.1\\beta) Q^{A_0 H}(2s', 1.1\\beta). \\] Then \\[ \\Theta_{s'} \\subset \\hat{\\Theta}_{s'}. \\]", "argument_gist": "Combines the two transverse exponentials by BCH and moves the small block-diagonal error into the remaining factors.", "gist_origin": "review_model", "proof_anchor": { "source": "manuscript", "line_start": 1321, "line_end": 1331, "char_start": 68171, "char_end": 68689 }, "proof_opening": "\\par Let us take $$g = a_0 a u^\\ast (r^\\ast) \\exp(\\vv w^-) \\exp(\\vv w^+) u(r) \\in \\Theta_{s'}.$$ By Lemma \\ref{lm:product-w}, \\[\\exp(\\vv w^-) \\exp(\\vv w^+) = \\exp(\\bar{\\vv w} ) \\exp( \\bar{\\vv h}), \\] where $\\bar{\\vv w} \\in B_{\\fr_1 + \\fr_2}(1.01\\beta e^{-s'})$ and $\\bar{\\vv h} \\in B_{\\R \\fa_0 + \\fh} (0.01\\beta e^{-2s'})$. Therefore, we have \\[ g = a_0 a u^\\ast (r^\\ast) \\exp(\\bar{\\vv w}) \\exp(\\bar{\\vv h}) u(r).\\] By rearranging the equation above, we have that $g \\in \\hat{\\Theta}_{s'}$.", "anchor": { "source": "manuscript", "line_start": 1314, "line_end": 1321, "char_start": 67979, "char_end": 68170 }, "review": { "verdict": "minor_issues", "issues": [ "The numerical radius 1.1 requires a coordinate-compatible norm and explicit control of the rearrangement.", "For a Euclidean norm, two orthogonal transverse vectors each at their allowed radius need not have sum of norm at most 1.01 times that radius. The intended box-norm convention should be stated." ], "reasoning": "The claimed scale orders follow from the bracket calculations. The literal constants are not justified for an arbitrary norm." } }, { "id": "lm:product-theta", "kind": "lemma", "label": "Lemma 8.2", "title": "", "section": "sec-kakeya-model", "statement": "\\par Let us denote \\[ \\bar{\\Theta}_{s'} := Q_{s'}(2.21\\beta) Q^{A_0 H}(2s', 2.21\\beta). \\] Then for any $\\epsilon_i \\in \\{\\pm 1\\}$ for $i=1,2$, \\[ \\Theta^{\\epsilon_1}_{s'}\\Theta^{\\epsilon_2}_{s'} \\subset \\bar{\\Theta}_{s'}. \\]", "argument_gist": "Uses the enlarged Theta coordinate description, combines transverse factors, and controls the stable block-diagonal coordinate under products and inverses.", "gist_origin": "review_model", "proof_anchor": { "source": "manuscript", "line_start": 1338, "line_end": 1354, "char_start": 68980, "char_end": 69985 }, "proof_opening": "\\par We will only prove \\[ \\Theta_{s'}^{-1} \\Theta_{s'} \\subset \\bar{\\Theta}_{s'}. \\] Other cases can be proved similarly. \\par By Lemma \\ref{lm:two-theta}, it suffices to show that \\[ \\hat{\\Theta}_{s'}^{-1} \\hat{\\Theta}_{s'} \\subset \\bar{\\Theta}_{s'},\\] where $\\hat{\\Theta}_{s'}$ is defined as in Lemma \\ref{lm:two-theta}. \\par Let us take $g = \\exp(\\vv w) \\exp(\\vv h) \\in \\hat{\\Theta}_{s'}$ and $g' = \\exp(\\vv w') \\exp(\\vv h') \\in \\hat{\\Theta}_{s'}$. Then \\[ g^{-1} g' = \\exp(-\\vv h) \\exp(-\\vv w) \\exp(\\vv w') \\exp(\\vv h'). \\] By Lemma \\ref{lm:product-w}, we have \\[ \\exp(-\\vv w) \\exp(\\vv w') = \\exp(\\bar{\\vv w}) \\exp (\\bar{\\vv h}), \\] where $\\bar{\\vv w} \\in B_{\\fr_1 + \\fr_2}(2.201\\beta e^{-s'})$,", "anchor": { "source": "manuscript", "line_start": 1331, "line_end": 1338, "char_start": 68690, "char_end": 68979 }, "review": { "verdict": "minor_issues", "issues": [ "The anisotropic estimate for the fu* coordinate after rearrangement is left implicit.", "The precise constant 2.21 depends on the unresolved norm convention in lm:two-theta." ], "reasoning": "The small-scale multiplication mechanism is consistent with the Lie brackets. The omitted estimates appear local and repairable, unlike the later global conditioning and incidence steps." } }, { "id": "cor:product-theta", "kind": "cor", "label": "Cor 8.3", "title": "", "section": "sec-kakeya-model", "statement": "\\par Let us denote \\[ \\tilde{\\Theta}_{s'} := Q_{s'}(4.5\\beta) Q^{A_0 H}(2s', 4.5\\beta). \\] Then for any $\\epsilon_i \\in \\{\\pm 1\\}$ for $i=1,2,3,4$, \\[ \\Theta^{\\epsilon_1}_{s'}\\Theta^{\\epsilon_2}_{s'} \\Theta^{\\epsilon_3}_{s'} \\Theta^{\\epsilon_4}_{s'} \\subset \\tilde{\\Theta}_{s'}. \\]", "argument_gist": "Applies the two-factor neighborhood estimate twice to obtain a four-factor estimate.", "gist_origin": "review_model", "proof_anchor": { "source": "manuscript", "line_start": 1362, "line_end": 1372, "char_start": 70408, "char_end": 71027 }, "proof_opening": "\\par We will only prove \\[\\Theta_{s'}\\Theta_{s'} \\Theta_{s'} \\Theta_{s'} \\subset \\tilde{\\Theta}_{s'}. \\] Other cases can be proved similarly. In fact, by Lemma \\ref{lm:product-theta}, we have \\[ \\Theta_{s'}\\Theta_{s'} \\subset \\bar{\\Theta}_{s'}, \\] where $\\bar{\\Theta}_{s'}$ is defined as in Lemma \\ref{lm:product-theta}. By repeating the proof of Lemma \\ref{lm:product-theta}, we get \\[ \\bar{\\Theta}_{s'} \\bar{\\Theta}_{s'} \\subset \\tilde{\\Theta}_{s'}. \\] Combining the two inclusions above, we conclude that \\[\\Theta_{s'}\\Theta_{s'} \\Theta_{s'} \\Theta_{s'} \\subset \\tilde{\\Theta}_{s'}. \\]", "anchor": { "source": "manuscript", "line_start": 1355, "line_end": 1362, "char_start": 70057, "char_end": 70407 }, "review": { "verdict": "minor_issues", "issues": [ "Inherits the norm and numerical-radius qualifications of lm:product-theta." ], "reasoning": "Once a uniform two-factor estimate at the enlarged radius is proved, the stated iteration follows." } }, { "id": "lm:different-coordinate-1", "kind": "lemma", "label": "Lemma 8.4", "title": "", "section": "sec-kakeya-model", "statement": "For $\\vv w^+ \\in \\vv r^+$ and $ \\vv w^- \\in \\vv r^-$ with $\\|\\vv w^+\\|, \\|\\vv w^-\\| \\le 10 \\beta e^{-\\ell}$, we have that \\[ \\exp(\\vv w^+ + \\vv w^-) = a_0 a u^{\\ast} (r^\\ast) \\exp(\\bar{\\vv w}^-) \\exp(\\bar{\\vv w}^+) u(r), \\] where $\\bar{\\vv w}^+ \\in \\vv r^+$ with $\\|\\bar{\\vv w}^+ - \\vv w^+\\| \\le 0.01\\beta e^{-3\\ell}$, $\\bar{\\vv w}^- \\in \\vv r^-$ with $\\|\\bar{\\vv w}^- - \\vv w^-\\| \\le 0.01\\beta e^{-3\\ell}$, $a_0 a \\in B^{A_0 A}(0.01\\beta e^{-2\\ell})$, and $r^\\ast, r \\in [0.01\\beta e^{-2\\ell}]$. \\par Similarly, we have \\[ \\exp(\\vv w^+ + \\vv w^-) = a_0 a u (r) \\exp(\\bar{\\vv w}^+) \\exp(\\bar{\\vv w}^-) u^\\ast(r^\\ast), \\] where $\\bar{\\vv w}^+ \\in \\vv r^+$ with $\\|\\bar{\\vv w}^+ - \\vv w^+\\| \\le 0.01\\beta e^{-3\\ell}$, $\\bar{\\vv w}^- \\in \\vv r^-$ with $\\|\\bar{\\vv w}^- - \\vv w^-\\| \\le 0.01\\beta e^{-3\\ell}$, $a_0 a \\in B^{A_0 A}(0.01\\beta e^{-2\\ell})$, and $r^\\ast, r \\in [0.01\\beta e^{-2\\ell}]$.", "argument_gist": "Uses local product coordinates and BCH to show that changing the order of transverse and block-diagonal factors introduces quadratic diagonal and cubic transverse errors.", "gist_origin": "review_model", "proof_anchor": { "source": "manuscript", "line_start": 1384, "line_end": 1411, "char_start": 72109, "char_end": 74072 }, "proof_opening": "\\par We only prove the first statement. The second statement can be proved using the same argument. \\par Let us write \\[ \\exp(\\vv w^+ + \\vv w^-) = a_0 a u^{\\ast} (r^\\ast) \\exp(\\bar{\\vv w}^-) \\exp(\\bar{\\vv w}^+) u(r), \\] where $a_0 a \\in B^{A_0A} (10\\beta e^{-\\ell})$, $r^\\ast, r \\in [10\\beta e^{-\\ell}]$, $\\bar{\\vv w}^- \\in B_{\\vv r^-}(10 \\beta e^{-\\ell})$, and $\\bar{\\vv w}^+ \\in B_{\\vv r^+}(10 \\beta e^{-\\ell})$. Noting that \\[ \\|[\\bar{\\vv w}^-, \\bar{\\vv w}^+]\\| \\le \\|\\bar{\\vv w}^-\\| \\cdot \\|\\bar{\\vv w}^+\\| \\le 0.001 \\beta e^{-2\\ell}, \\] by Lemma \\ref{lm:product-w}, we have \\[ \\exp(\\bar{\\vv w}^-) \\exp(\\bar{\\vv w}^+) = \\exp(\\tilde{\\vv w}^- + \\tilde{\\vv w}^+) \\exp( \\vv h), \\] where $\\tilde{\\vv w", "anchor": { "source": "manuscript", "line_start": 1373, "line_end": 1384, "char_start": 71114, "char_end": 72108 }, "review": { "verdict": "minor_issues", "issues": [ "The initial product-coordinate existence and uniqueness are assumed without an inverse-function argument.", "The exact small constants need a fixed norm and a sufficiently small beta.", "The lemma should explicitly specify the intended nonnegative range of ell and the smallness assumptions on beta." ], "reasoning": "The quadratic and cubic orders are supported by the symmetric transverse/block-diagonal bracket decomposition. The remaining concerns are local quantitative details." } }, { "id": "lm:cross-section", "kind": "lemma", "label": "Lemma 8.5", "title": "", "section": "sec-kakeya-model", "statement": "For $y \\in \\Pi(x)$ and $r \\in [\\beta e^{2s'}]$, let \\[J_r(y) :=\\left\\{r' \\in \\left[2\\beta e^{2s'}\\right]: u(r')y \\in \\Sigma^r\\right\\}. \\] Then there exists intervals $J_1 \\subset J_r(y) \\subset J_2$ with size $ 0.5 \\beta \\le |J_1| \\le |J_2| \\le 4 \\beta$.", "argument_gist": "Uses the explicit orbit-coordinate formulas to solve for a prescribed U-coordinate and to estimate the time spent in a cross-section.", "gist_origin": "review_model", "proof_anchor": { "source": "manuscript", "line_start": 1540, "line_end": 1568, "char_start": 82013, "char_end": 84805 }, "proof_opening": "\\par Without loss of generality, we can assume $r=0$. We first prove $J_0(y) \\neq \\emptyset$. For any $y \\in \\Pi(x)$, let us write \\[ y = a_y a_{0, y} u^\\ast(r^\\ast_y) \\exp(\\vv w_y^-) \\exp(\\vv w_y^+) u(r_y) x \\] where $a_y a_{0,y} \\in B^{A A_0} (\\beta)$, $r_y^\\ast \\in \\left[\\beta e^{-2s'}\\right]$, $$\\vv w_y^- = v_{2,y} \\vv v_2 + w_{2,y} \\vv w_2 \\in B_{\\vv r^-} \\left(\\beta e^{-s'}\\right),$$ $$\\vv w_y^+ = v_{1,y} \\vv v_1 + w_{1,y} \\vv w_1 \\in B_{\\vv r^+} \\left(\\beta e^{s'}\\right),$$ and $r_y \\in \\left[\\beta e^{2s'}\\right]$. Then for any $r \\in \\left[\\beta e^{2s'}\\right]$, \\[ u(r) y = a'_y a'_{0, y} u^{\\ast} (\\tilde r^\\ast_y) u(\\tilde r) \\exp(\\vv w_y^-) \\exp(\\vv w_y^+) u(r_y) x,\\] where $a'_y a", "anchor": { "source": "manuscript", "line_start": 1414, "line_end": 1421, "char_start": 74296, "char_end": 74614 }, "review": { "verdict": "major_issues", "issues": [ "Solving f_y(tilde r)=0 controls only the projected U-coordinate. The proof does not show that the other coordinates remain within the defining bounds of Sigma^r.", "The claim Sigma(u(r)x) subset Pi(x) is used without boundary margins; translating a cross-section at an endpoint can exceed the playground's U-range.", "The reduction to r=0 and the sign-change argument must preserve the permitted time interval. This is not checked.", "The lower interval inclusion requires the transverse and diagonal coordinates to remain admissible throughout that interval, not merely |f_z(r)|<=beta." ], "reasoning": "The projected time coordinate has the expected derivative near one. That fact alone does not establish membership in the full anisotropic neighborhood. This is an unverified geometric step, not a counterexample to a buffered formulation." } }, { "id": "lm:initial-dimension-under-a-t", "kind": "lemma", "label": "Lemma 8.6", "title": "", "section": "sec-kakeya-model", "statement": "For any $\\delta_3 t \\le \\ell \\le \\delta_1 t$ and any $\\bar x \\in \\Pi(x)$, we have \\[ \\tilde \\mu(\\Sigma_{s', \\ell} (\\bar x)) \\le e^{-2s'} e^{-d_0 \\ell}.\\]", "argument_gist": "Pulls an anisotropic cross-section back by a(-2s'), applies initial dimension control, and uses its short U-width for an additional e^{-2s'} factor.", "gist_origin": "review_model", "proof_anchor": { "source": "manuscript", "line_start": 1433, "line_end": 1448, "char_start": 75276, "char_end": 76411 }, "proof_opening": "\\par Note that $\\Sigma_{s', \\ell} (\\bar x)$ is a cross-section of $\\Pi_{s', \\ell}(\\bar x)$ along the $U$-orbits, and $\\tilde \\mu$ is Lebesgue along the $U$-orbits. We have \\[ \\tilde \\mu(\\Sigma_{s', \\ell} (\\bar x) ) \\ll e^{-2s'} \\tilde \\mu (\\Pi_{s', \\ell} (\\bar x) ).\\] Note that by the definition of $\\tilde \\mu$, we have \\[ \\tilde \\mu (\\Pi_{s', \\ell} (\\bar x)) \\le \\mu_{\\cF_i} (a(-2s')\\Pi_{s', \\ell} (\\bar x) ), \\] and \\[ a(-2s')\\Pi_{s', \\ell} (\\bar x) = B^{A_0 A}(\\beta) B^{U^\\ast}(\\beta) \\exp\\left( B_{\\vv r^-}(\\beta e^{-\\ell}) \\right) \\exp\\left( B_{\\vv r^+}(\\beta e^{-\\ell}) \\right) B^{U}(\\beta) x' \\] where $x' = a(-2s') \\bar x$. By Lemma \\ref{lm:different-coordinate-1}, we have that \\[ B^{A_0 ", "anchor": { "source": "manuscript", "line_start": 1428, "line_end": 1433, "char_start": 75050, "char_end": 75275 }, "review": { "verdict": "major_issues", "issues": [ "The conditioning inequality is incorrect. For A contained in a full playground, tilde mu(A)=mu(A)/mu(Pi(x)); one cannot bound it by the unconditioned pulled-back measure without the factor mu(Pi(x))^{-1}, potentially beta^{-15}.", "The measure has been restricted by exceptional-set removal; being a restriction of orbit Lebesgue measure does not automatically make it uniform along complete U-segments.", "The proof uses initial dimension control at radius 2beta and arbitrary centers, beyond the literal statement proved for radius beta and centers in F." ], "reasoning": "The missing conditioning denominator is an explicit measure-theoretic error. The later strange-part proof itself reintroduces a beta^{-15} factor, showing the distinction matters." } }, { "id": "def:kakeya-model", "kind": "defn", "label": "Defn 8.7", "title": "", "section": "sec-kakeya-model", "statement": "\\par Let $s'>0$ and $\\tilde \\mu$ be a probability measure on a playground $\\Pi(x) \\subset X$ defined by taking the normalized measure on a subset consisting of $U$-orbits of length $\\beta e^{2s'}$. Let us fix a sequence of cross-sections $$\\left\\{\\Sigma^r = \\Sigma_{s'} u(r)x: r \\in \\left[\\beta e^{2s'}\\right]\\cap\\left(0.1\\beta \\Z\\right) \\right\\} .$$ \\par We first define the projection map \\[ \\bP: \\Pi(x) \\to \\R^3 \\] as follows: For $x' \\in \\Pi(x)$, we can write \\[x' =a_{0, x'} a_{x'} u^\\ast(r^\\ast_{x'} ) \\exp(\\vv w^-_{x'}) \\exp(\\vv w^+_{x'}) u(r_{x'} ) x\\] where $a_{0, x'} \\in B^{A_0}(\\beta)$, $a_{x'} \\in B^A(\\beta)$, $\\vv w^-_{x'} = v_{2, x'} \\vv v_2 + w_{2, x'} \\vv w_2 \\in \\vv r^- $ and $\\vv w^+_{x'} = v_{1, x'} \\vv v_1 + w_{1, x'} \\vv w_1 \\in \\vv r^+$, $ r^{\\ast}_{x'} \\in [\\beta e^{-2s'}]$, and $r_{x'}\\in [\\beta e^{2s'}]$. Then $$\\bP(x') := (r_{x'}, v_{1,x'}, w_{1, x'}).$$ Then $$\\bP(\\Pi(x)) = \\left[\\beta e^{2s'}\\right] \\times \\left[\\beta e^{s'}\\right]^2 \\subset \\R^3.$$ It is easy to see that for $r \\in \\left[\\beta e^{2s'}\\right]$, \\[ \\bP(\\Sigma^r) = (r, 0, 0) + [\\beta] \\times \\left[\\beta e^{s'}\\right]^2. \\] \\par Let $\\cI$ be a finite subset of $\\Pi(x)$ defined as follows: \\cI := \\left\\{ \\exp(v_1 \\fd \\vv v_1 + w_1 \\fd \\vv w_1 ) u(r \\fd') x: r, v_1, w_1 \\in \\left[10 e^{2s'}\\right]\\cap \\Z \\right\\}, where $\\fd = 0.1 \\beta e^{-s'}, \\fd' = 0.1 \\beta$. For $z \\in \\cI$, let $\\Theta'(z) := \\Theta(z) \\cap \\Pi(x)$. For every $z \\in \\cI$, let $\\cI_z$ be a finite subset of $\\Theta'(z)$ defined as follows: \\cI_z := \\left\\{ \\exp(v_2 \\fd'' \\vv v_2 + w_2 \\fd'' \\vv w_2 ) z : v_2, w_2 \\in \\left[10 e^{2s'}\\right]\\cap \\Z \\right\\}, where $\\fd'' = 0.1 \\beta e^{-3s'}$. For $y \\in \\cI_z$, let $\\Omega'(y):= \\Omega(y) \\cap \\Theta'(z)$. Let us denote $$\\fJ := \\bigcup_{z \\in \\cI} \\cI_z.$$ \\par For any $y \\in \\Pi(x)$, let us define \\fl(y) := \\left\\{ r \\in \\left[2\\beta e^{2s'}\\right]: u(r) y \\in \\Pi(x) \\right\\}. \\par For each $z \\in \\cI$ and $y \\in \\cI_z$, let us write \\[ y = \\exp\\left(\\vv w^-_y\\right) \\exp\\left(\\vv w^+_y\\right) u\\left(r_y \\right) x,\\] where $$\\vv w^-_y = v_{2, y} \\vv v_2 + w_{2, y} \\vv w_2 \\in B_{\\vv r^-}\\left(\\beta e^{-s'}\\right),$$ $$\\vv w^+_y = v_{1, y} \\vv v_1 + w_{1, y} \\vv w_1 \\in B_{\\vv r^+}\\left(\\beta e^{s'}\\right),$$ and $r_y \\in \\left[\\beta e^{2s'}\\right]$. Let us define the curve $\\cL(y)$ as follows: \\[ \\cL(y) := \\left\\{ \\left(f_y(r), f_{v, y}(r), f_{w, y}(r) \\right) = \\bP\\left(u(r) y\\right) : r \\in \\fl(y) \\right\\}. \\] It is straightforward to check that f_{v, y}(r) = \\frac{2r v_{2, y}}{2 + r v_{2, y} w_{2, y}} + v_{1, y}, f_{w, y}(r) = \\frac{-2r w_{2, y}}{2 - r v_{2, y} w_{2, y}} + w_{1, y}, and {rl} f_y(r) & = r_y + \\frac{2r + 2r w_{1,y} v_{2, y}}{2 + r v_{2, y} w_{2, y}} + \\frac{w_{1,y}v_{1,y} - f_{v, y}(r)f_{w, y}(r)}{2} \\\\ &= r_y + r + \\frac{r w_{2,y} v_{1,y}}{2 - r w_{2,y} v_{2,y}} + \\frac{r w_{1,y} v_{2,y}}{2 + r w_{2,y} v_{2,y}} + \\frac{ r (r w_{2,y} v_{2,y})^2 }{4 - (r w_{2,y} v_{2,y})^2} \\par From \\eqref{eq:fy} it is easy to see that $|f_{y} (r) - r_y - r| \\le \\beta |r|$. \\par Let $\\cT(y)$ denote the $2\\beta e^{-s'}$-neighborhood of $\\cL(y)$. We then assign to $\\cT(y)$ a weight \\[\\cW(y) = \\tilde{\\mu}(\\Omega'(y)).\\] \\par For $y_1, y_2 \\in \\Pi(x)$, let us define $\\Delta(y_1, y_2)$ as follows: Let us write \\[ y_2 = a a_0 u^\\ast(r^\\ast) \\exp(\\vv w^-) \\exp(\\vv w^+) u(r) y_1,\\] where $a a_0 \\in B^{A A_0} (2\\beta)$, $r^\\ast \\in \\left[2\\beta e^{-2s'}\\right]$, $\\vv w^- = v_2 \\vv v_2 + w_2 \\vv w_2 \\in B_{\\vv r^-} \\left(2\\beta e^{-s'}\\right)$, and $\\vv w^+ = v_1 \\vv v_1 + w_1 \\vv w_1 \\in B_{\\vv r^+} \\left(2\\beta e^{s'}\\right)$. Then we define \\Delta(y_1, y_2) = (v_2, w_2 ). Let us call the collection of tubes $\\{\\cT(y): y \\in \\fJ\\}$ with weights $\\{\\cW(y)\\}$ the Kakeya-type model for $(\\tilde \\mu, \\Pi(x))$.", "argument_gist": "\\par Let $s'>0$ and $\\tilde \\mu$ be a probability measure on a playground $\\Pi(x) \\subset X$ defined by taking the normalized measure on a subset consisting of $U$-orbits of length $\\beta e^{2s'}$. Let us fix a sequence of cross-sections $$\\left\\{\\Sigma^r = \\Sigma_{s'} u(r)x: r \\in \\left[\\beta e^{2s'}\\right]\\cap\\left(0.1\\beta \\Z\\right) \\right\\} .$$ \\par We first define the projection map \\[ \\bP: \\Pi(x) \\to \\R^3 \\] as follows: For $x' \\in \\Pi(x)$, we can write \\[x' =a_{0, x'} a_{x'} u^\\ast(r^\\ast_{x'} ) \\exp(\\vv w^-_{x'}) \\exp(\\vv w^+_{x'}) u(r_", "gist_origin": "extractive_statement", "anchor": { "source": "manuscript", "line_start": 1451, "line_end": 1538, "char_start": 76498, "char_end": 81959 } }, { "id": "lm:partition-1", "kind": "lemma", "label": "Lemma 8.8", "title": "", "section": "sec-kakeya-model", "statement": "We have \\[ \\Pi(x) = \\bigcup_{z \\in \\cI} \\Theta'(z). \\] Moreover, for any $z' \\in \\Pi(x)$, there are at most $N_2$ different $z \\in \\cI$ such that \\[ \\Theta(z') \\cap \\Theta(z) \\neq \\emptyset. \\]", "argument_gist": "Rounds the three projected coordinates to a grid and uses neighborhood product estimates to bound overlap multiplicity.", "gist_origin": "review_model", "proof_anchor": { "source": "manuscript", "line_start": 1619, "line_end": 1660, "char_start": 87795, "char_end": 90925 }, "proof_opening": "\\par We first prove that for any $z' \\in \\Pi(x)$, there exists $z \\in \\cI$ such that $z' \\in \\Theta(z)$. In fact, let us write \\[ z' = a_{z'} a_{0, z'} u^{\\ast} (r^\\ast_{z'}) \\exp(\\vv w^-_{z'}) \\exp(v_{1, z'} \\vv v_1 + w_{1, z'} \\vv w_1) u(r_{z'}) x, \\] where $a_{z'} \\in B^A(\\beta)$, $a_{0, z'} \\in B^{A_0}(\\beta)$, $r^\\ast_{z'} \\in \\left[\\beta e^{-2s'}\\right]$, $\\vv w^-_{z'} \\in B_{\\vv r^-}\\left(\\beta e^{-s'}\\right)$, and $$(r_{z'} , v_{1, z'}, w_{1, z'}) \\in \\left[\\beta e^{2s'}\\right] \\times \\left[\\beta e^{s'}\\right]^2.$$ Let us choose $z \\in \\cI$ such that \\[ z = \\exp(v_1 \\vv v_1 + w_1 \\vv w_1 ) u(r_z) x \\] such that $|r_z - r_{z'}| \\le 0.2 \\beta$, and \\[ \\max \\{ |v_{1, z} - v_1|, |w_{1,z}", "anchor": { "source": "manuscript", "line_start": 1612, "line_end": 1619, "char_start": 87550, "char_end": 87794 }, "review": { "verdict": "major_issues", "issues": [ "The proof does not establish the global uniqueness of the playground coordinates used by the projection P.", "The inference from eq:z-z' to a small axis-aligned difference P(z)-P(z') ignores the Heisenberg cross term. A transverse increment of size beta e^{-s'} against a coordinate of size beta e^{s'} contributes a U-coordinate shear of order beta^2, not beta e^{-s'}. A sheared-box counting argument is needed.", "Grid endpoints and the use of balls versus the declared centered intervals are not consistently controlled." ], "reasoning": "A bounded-overlap cover may be obtainable in suitable sheared coordinates. The displayed coordinate-difference justification does not prove the asserted multiplicity." } }, { "id": "lm:partition-2", "kind": "lemma", "label": "Lemma 8.9", "title": "", "section": "sec-kakeya-model", "statement": "For any $z \\in \\cI$, \\[ \\Theta'(z) = \\bigcup_{y \\in \\cI_z} \\Omega'(y). \\] Moreover, for any $y' \\in \\Theta(z)$, there are at most $N_2$ different $y \\in \\cI_z$ such that \\[ \\Omega(y) \\cap \\Omega(y') \\neq \\emptyset. \\]", "argument_gist": "Rounds the two contracting transverse coordinates on a finer grid and uses product estimates to obtain an Omega cover with bounded overlap.", "gist_origin": "review_model", "proof_anchor": { "source": "manuscript", "line_start": 1676, "line_end": 1727, "char_start": 91776, "char_end": 95532 }, "proof_opening": "\\par We first prove that for any $z' \\in \\Theta'(z)$, there exists $y \\in \\cI_z$ such that $z' \\in \\Omega(y)$. By Lemma \\ref{lm:two-theta}, $\\Theta_{s'} \\subset \\hat{\\Theta}_{s'}$. Therefore, we can write \\[ z' = a_0 a u(r) \\exp(\\vv w^+ + \\vv w^-)u^\\ast (r^\\ast) z, \\] where $a_0 a \\in B^{A_0 A} (1.1 \\beta)$, $r \\in [1.1\\beta]$, $r^\\ast \\in [1.1 \\beta e^{-2s'}]$, $\\vv w^+ \\in B_{\\vv r^+}(1.1 \\beta e^{-s'})$ and $\\vv w^- = v_2 \\vv v_2 + w_2 \\vv w_2 \\in B_{\\vv r^-}(1.1\\beta e^{-s'})$. Let $y \\in \\cI_z$ such that \\[ y = \\exp(\\bar{\\vv w}^-) z, \\] with $\\| \\bar{\\vv w}^- - \\vv w^- \\|\\le 0.2 \\beta e^{-3s'}$. \\par We will prove that $z' \\in \\Omega(y)$. In fact, we have z' = a_0 a u(r) \\exp(\\vv w^+ + ", "anchor": { "source": "manuscript", "line_start": 1660, "line_end": 1667, "char_start": 90926, "char_end": 91197 }, "review": { "verdict": "major_issues", "issues": [ "The final equality between the coordinate difference D_z(z')-D_z(y) and the relative transverse coordinate requires a coordinate-composition calculation; it is not a consequence of the preceding factorization alone.", "The grid range, the assertion I_z subset Theta'(z), and boundary coverage are not reconciled with the enlarged radii used in the proof.", "The use of local coordinates on X inherits the unestablished injectivity assumptions in the playground construction." ], "reasoning": "The fine-scale covering idea is reasonable, but the bounded-overlap statement is indispensable for the weighted incidence sums and has not been rigorously derived." } }, { "id": "def:coordinate-theta", "kind": "defn", "label": "Defn 8.10", "title": "", "section": "sec-kakeya-model", "statement": "\\par Given $z \\in \\cI$ and $ y \\in \\Theta'(z)$, let us define the coordinate of $y$ in $\\Theta'(z)$, denoted by $\\fD_z(y) \\in \\R^2$ as follows: Let us write \\[ y = a_0 a u(r_y) \\exp(\\vv w^+) \\exp( \\vv w^-)u^\\ast (r^\\ast_y) z, \\] where $a_0 a \\in A_0 A$, $\\vv w^+ \\in \\vv r^+$ and $\\vv w^- = v_2 \\vv v_2 + w_2 \\vv w_2 \\in \\vv r^-$. Then we define \\[ \\fD_z(y) := (v_2, w_2). \\]", "argument_gist": "\\par Given $z \\in \\cI$ and $ y \\in \\Theta'(z)$, let us define the coordinate of $y$ in $\\Theta'(z)$, denoted by $\\fD_z(y) \\in \\R^2$ as follows: Let us write \\[ y = a_0 a u(r_y) \\exp(\\vv w^+) \\exp( \\vv w^-)u^\\ast (r^\\ast_y) z, \\] where $a_0 a \\in A_0 A$, $\\vv w^+ \\in \\vv r^+$ and $\\vv w^- = v_2 \\vv v_2 + w_2 \\vv w_2 \\in \\vv r^-$. Then we define \\[ \\fD_z(y) := (v_2, w_2). \\]", "gist_origin": "extractive_statement", "anchor": { "source": "manuscript", "line_start": 1668, "line_end": 1675, "char_start": 91299, "char_end": 91729 } }, { "id": "rmk:partition-2", "kind": "remark", "label": "Remark 8.11", "title": "", "section": "sec-kakeya-model", "statement": "\\par Let us denote \\fC := \\left[1.2 \\beta e^{-s'}\\right]^2. From Lemma \\ref{lm:partition-2}, we can see that for any $z' \\in \\Theta(z)$, \\[ \\fD_z(z') \\in \\fC. \\]", "argument_gist": "\\par Let us denote \\fC := \\left[1.2 \\beta e^{-s'}\\right]^2. From Lemma \\ref{lm:partition-2}, we can see that for any $z' \\in \\Theta(z)$, \\[ \\fD_z(z') \\in \\fC. \\]", "gist_origin": "extractive_statement", "anchor": { "source": "manuscript", "line_start": 1727, "line_end": 1736, "char_start": 95533, "char_end": 95821 } }, { "id": "lm:wv2r", "kind": "lemma", "label": "Lemma 8.12", "title": "", "section": "sec-kakeya-model", "statement": "\\par Suppose \\[ y_2 = \\exp(\\vv w^-) \\exp(\\vv w^+) u(r_0) y_1,\\] where $|r_0| \\le \\beta e^{2s'}$, $\\vv w^- = v_2 \\vv v_2 + w_2 \\vv w_2 \\in B_{\\vv r^-} (\\beta e^{-s'})$, and $\\vv w^+ = v_1 \\vv v_1 + w_1 \\vv w_1 \\in B_{\\vv r^+} (\\beta e^{s'})$. For $r \\in [\\beta e^{2s'}]$, let us write $u(r) y_2$ as \\[ u(r) y_2 = a a_0 u^\\ast \\exp(\\vv w^-(r)) \\exp(\\vv w^+(r)) u(f(r)) y_1, \\] where $a \\in A$, $a_0 \\in A_0$, $u^\\ast \\in U^\\ast$, $\\vv w^+ (r) \\in \\vv r^+$, and \\[ \\vv w^-(r) = v_2(r) \\vv v_2 + w_2 (r) \\vv w_2 \\in \\vv r^-. \\] Then v_2(r) = \\frac{2 v_2}{2 - r w_2 v_2}, and w_2(r) = \\frac{2 w_2}{2 + r w_2 v_2}. In particular, for any $r' \\in [\\beta e^{2s'}]$, if we denote $y'_1 = u(r') y_1$, $y'_2 = u(r) y_2$, we have \\[ \\|\\Delta(y'_1, y'_2)\\| \\asymp \\|\\Delta(y_1, y_2)\\|. \\]", "argument_gist": "Computes the two contracting transverse coordinates under U using the ordered matrix factorization.", "gist_origin": "review_model", "proof_anchor": { "source": "manuscript", "line_start": 1758, "line_end": 1761, "char_start": 96790, "char_end": 96861 }, "proof_opening": "This can be verified by direct calculation.", "anchor": { "source": "manuscript", "line_start": 1736, "line_end": 1758, "char_start": 95822, "char_end": 96789 }, "review": { "verdict": "minor_issues", "issues": [ "The concluding comparison for Delta requires the relevant relative-coordinate chart to be defined; this qualification is missing for the stated points.", "Small beta is required to keep 2 plus or minus r w_2 v_2 bounded away from zero." ], "reasoning": "The rational formulas have the asserted scale behavior because |r w_2 v_2| is O(beta^3). Within the intended small injective chart, the comparability follows." } }, { "id": "lm:omega-u", "kind": "lemma", "label": "Lemma 8.13", "title": "", "section": "sec-kakeya-model", "statement": "Let $\\tilde \\mu$ and $\\Pi(x)$ be as in Definition \\ref{def:kakeya-model}. Let us denote \\hat{\\Omega}_{s'} := Q^{s'}_{3s'}(2.1\\beta) Q^{A_0 H} (2s', 2.1\\beta). For $y \\in \\fJ$, let us denote U\\Omega'(y) : = \\left\\{ u(r') y' : y' \\in \\Omega'(y), r' \\in \\fl(y') \\right\\}. For $r \\in \\fl(y)$, let us denote \\Omega(y, r) := U\\Omega'(y) \\cap \\Sigma^{\\tilde{r}}, where $\\tilde r$ satisfies that $u(r)y \\in \\Sigma^{\\tilde{r}}$. Then \\[ \\Omega(y, r) \\subset \\hat{\\Omega}(u(r)y) := \\hat{\\Omega}_{s'} u(r) y. \\] In particular, we have \\[ \\tilde \\mu(\\Omega'(y)) \\ll \\tilde \\mu \\left(\\hat{\\Omega} (u(r) y)\\right). \\]", "argument_gist": "Uses the shearing relation Ad(u(r))r^- subset r^-+r^+ and an H time change to transport Omega packets between cross-sections.", "gist_origin": "review_model", "proof_anchor": { "source": "manuscript", "line_start": 1782, "line_end": 1804, "char_start": 97808, "char_end": 99380 }, "proof_opening": "\\par For $y' \\in \\Omega'(y)$, we have \\[ y = \\exp(\\vv w^+ + \\vv w^-) \\exp(\\vv h) y', \\] where $\\vv w^+ \\in B_{\\vv r^+}(2.01\\beta e^{-s'})$, $\\vv w^- \\in B_{\\vv r^-}(2.01\\beta e^{-3s'})$, and $\\vv h \\in \\fq^{A_0 H}(2s', 2\\beta)$. Then u(r) y &= u(r) \\exp(\\vv w^+ + \\vv w^-) \\exp(\\vv h) y' \\\\ &= \\exp(\\Ad_{u(r)} (\\vv w^+ + \\vv w^-)) u(r)\\exp(\\vv h) y' \\\\ &= \\exp(\\vv w^+ + \\tilde{\\vv w}^+ + \\vv w^-) \\exp(\\tilde{\\vv h}) u(r') y', where $\\tilde{\\vv w}^+ \\in \\vv r^+$ with $\\|\\tilde{\\vv w}^+\\| = |r| \\|\\vv w^-\\|$, and $\\tilde{\\vv h} \\in \\fq^{A_0 H}(2s', 2.01\\beta)$. Note that $|r| \\le \\beta e^{2s'}$, and $\\|\\vv w^-\\| \\le 2.01 \\beta e^{-3s'}$. We have $\\|\\tilde{\\vv w}^+\\| \\le 0.01 \\beta e^{-s'}$. This ", "anchor": { "source": "manuscript", "line_start": 1763, "line_end": 1782, "char_start": 96976, "char_end": 97807 }, "review": { "verdict": "major_issues", "issues": [ "The set inclusion depends on unproved cross-section boundary control.", "The implication from packet transport to comparable tilde mu masses assumes uniform conditional orbit density and comparable complete orbit lengths after restriction; these hypotheses have not been established.", "The proof starts with the inverse relation for y in terms of y', although membership was defined in the opposite direction, and needs anisotropic inverse estimates with the stated constants.", "A time-change estimate for derivatives or Jacobians is needed for the mass comparison, not only proximity of the times." ], "reasoning": "The Lie algebra shear has the correct scales. The geometric and measure-theoretic conclusions require additional arguments." } }, { "id": "lm:time-leave", "kind": "lemma", "label": "Lemma 8.14", "title": "", "section": "sec-kakeya-model", "statement": "\\par Let $s' >0$ and $\\Pi(x) $ be as in Definition \\ref{def:kakeya-model}. Let us fix a cross-section $\\Sigma^r \\subset \\Pi(x)$ defined as above. Then for any $y_1, y_2 \\in \\fJ \\cap \\Sigma^r$, suppose there exist $L_1, L_2$ and $z \\in \\cI$ such that \\[ \\Omega(y_i, L_i) \\cap \\Theta'(z) \\neq \\emptyset, \\text{ for } i=1,2, \\] where $\\Omega(y_i, L_i)$ is defined as in \\eqref{eq:omega-y-r}. \\par Let us denote $\\tilde{y}_i = u(L_i) y_i$ and write \\[ \\tilde{y}_1 = a_0 a u^\\ast(r^\\ast) \\exp(\\vv w^-) \\exp(\\vv w^+) u(r_0) \\tilde{y}_2,\\] where $a_0 a \\in A_0 A$, $$\\vv w^- = v_2 \\vv v_2 + w_2 \\vv w_2 \\in \\vv r^-,$$ and $$\\vv w^+ = v_1 \\vv v_1 + w_1 \\vv w_1 \\in \\vv r^+.$$ Then $\\vv w^+ \\in B_{\\vv r^+}(4.6 \\beta e^{-s'})$. Moreover, let I (y_1, y_2) := \\left\\{ r_1 \\in \\fl(y_1) : \\exists r_2 , z' \\in \\cI, \\text{ s.t. } \\Omega (y_i, r_i) \\cap \\Theta'(z')\\neq \\emptyset, i=1,2 \\right\\}. Then there exists an interval $I$, such that $ I(y_1, y_2) \\subset I$ with $$|I| \\asymp \\min\\left\\{ \\beta e^{2s'}, \\beta e^{-s'} \\|\\vv w^-\\|^{-1}\\right\\}.$$", "argument_gist": "Bounds how long two packets can remain in a common Theta neighborhood by comparing expanding transverse coordinates and their relative contracting slope.", "gist_origin": "review_model", "proof_anchor": { "source": "manuscript", "line_start": 1822, "line_end": 1859, "char_start": 100557, "char_end": 103129 }, "proof_opening": "\\par By Lemma \\ref{lm:omega-u}, we have $\\Omega(y_i, L_i) \\subset \\hat{\\Omega}(\\tilde{y}_i)$. Therefore, we have $ \\hat{\\Omega}(\\tilde{y}_1) \\cap \\Theta(z) \\neq \\emptyset$ which implies that \\[ \\tilde{y}_1 \\in \\hat{\\Omega}_{s'}^{-1} \\Theta_{s'} z \\subset \\Theta_{s'}^{-1} \\Theta_{s'} z. \\] Similarly, we have $ \\hat{\\Omega}(\\tilde{y}_2) \\cap \\Theta(z) \\neq \\emptyset$ implies that \\[ z \\in \\Theta_{s'}^{-1} \\Omega_{s'} \\tilde{y}_2 \\subset \\Theta_{s'}^{-1} \\Theta_{s'} \\tilde{y}_2. \\] Therefore, \\[ \\tilde{y}_1 \\in \\Theta_{s'}^{-1} \\Theta_{s'}\\Theta_{s'}^{-1} \\Theta_{s'} \\tilde{y}_2. \\] By Corollary \\ref{cor:product-theta}, \\[ \\tilde{y}_1 \\in \\tilde{\\Theta}_{s'} \\tilde{y}_2.\\] Then we have \\[ \\tild", "anchor": { "source": "manuscript", "line_start": 1804, "line_end": 1822, "char_start": 99381, "char_end": 100556 }, "review": { "verdict": "major_issues", "issues": [ "The proof treats hat Omega as contained in Theta, although hat Omega has radii 2.1beta and Theta has radius beta in shared coordinates. The resulting four-Theta containment and constant 4.6 do not follow.", "The conversion from increments based at the tilde y_i to the originally defined I(y_1,y_2) needs explicit time shifts and time-change control.", "The relative-coordinate and packet-transport prerequisites remain unproved." ], "reasoning": "The rational expanding-coordinate formulas support an upper residence-time estimate in a correctly buffered chart. The supplied neighborhood inclusions do not establish the stated version." } }, { "id": "lm:concentrated-surface", "kind": "lemma", "label": "Lemma 8.15", "title": "", "section": "sec-kakeya-model", "statement": "\\par Recall that $I(y_1, y_2)$ is defined as in \\eqref{eq:Iy1y2}. Let us fix a cross-section $\\Sigma^r$. Let $z \\in \\cI\\cap \\Sigma^r, y \\in \\fJ\\cap \\Sigma^r$. Then the set of all coordinates $\\fD_z (\\bar z)$ of the elements $\\bar z \\in \\cI_z$ such that $I(\\bar z , y) \\neq \\emptyset$ and in addition there is some $r \\in I(\\bar z, y)$ with $|r| \\ge \\beta^{100} e^{2s'}$ is contained in the $ \\beta^{-100} e^{-3s'}$ neighborhood of the hyperbola \\left\\{ (X, Y) \\in \\fC: (X - X_0)(Y-Y_0) + \\frac{X - X_0}{v_1} + \\frac{Y - Y_0}{w_1} = 0 \\right\\}, where $v_1, w_1, X_0, Y_0$ are constants depending on $y, z$ with bounds $(X_0, Y_0) \\in \\fC$ and $(v_1, w_1) \\in [\\beta e^{s'}]^2$. The set of such coordinates $\\fD_z(\\bar z)$ is also contained in the box \\fB(\\cT(y), z) := (X_0, Y_0) + \\left[\\beta^{-100} e^{-2s'} \\|(v_1, w_1)\\|\\right]^2. Moreover, if there exists $\\bar z \\in \\cI_z$ such that there is some $r \\in I(\\bar z, y)$ with $|r| \\ge \\beta^{100} e^{2s'}$ and \\[ \\|\\Delta(\\bar z , y)\\| \\le \\beta e^{-s'-j}, \\] then \\[ \\|(v_1, w_1)\\| \\le \\beta^2 e^{s' -j},\\] and the size of $\\fB(\\cT(y), z)$ is $\\le \\beta^{-98} e^{s'-j}$.", "argument_gist": "Eliminates the encounter time from two rational transverse-coordinate equations to place interacting packet coordinates near a hyperbola and inside a small box.", "gist_origin": "review_model", "proof_anchor": { "source": "manuscript", "line_start": 1881, "line_end": 1934, "char_start": 104826, "char_end": 107929 }, "proof_opening": "\\par There exist $y' \\in \\Omega(y)$ and $z' \\in \\Theta(z)$ such that \\[ z' = \\exp(\\vv w^+) u(r_{y'}) y',\\] where $\\vv w^+ = v_1 \\vv v_1 + w_1 \\vv w_1 $, and \\[ z' = \\exp(\\vv w_0^-) z, \\] where $\\vv w_0^- = X_0 \\vv v_2 + Y_0 \\vv w_2$. We can also make sure that \\[ \\min\\{|v_1|, |w_1|\\} \\ge \\beta e^{-s'}. \\] \\par Without loss of generality, let us assume $|w_1| \\ge |v_1|$. The opposite case can be handled similarly. \\par Suppose $\\bar z = \\exp(\\vv w^-) z' $ (where $\\vv w^- = v_2 \\vv v_2 + w_2 \\vv w_2$) satisfies that $I(\\bar z, y) \\neq \\emptyset$. Then by \\eqref{eq:fvy}, \\eqref{eq:fwy}, and \\eqref{eq:fy}, we have that \\[ u(r) \\bar z = a_{0,r} a_r u^{\\ast} (f^\\ast(r)) \\exp(\\vv f_-(r)) \\exp(\\vv f", "anchor": { "source": "manuscript", "line_start": 1862, "line_end": 1880, "char_start": 103494, "char_end": 104824 }, "review": { "verdict": "major_issues", "issues": [ "The existence of the selected y', z' and nonzero v_1,w_1 with the required uniform lower bounds is asserted without proof.", "The division leading to eq:2 and eq:1 is unjustified when |v_1| or |w_1| is only beta e^{-s'}, while the permitted residual is 4.6beta e^{-s'}. That residual can be larger than the quantity being inverted.", "For example, v_2=0 and v_1=beta e^{-s'} satisfy the residual inequality, but the claimed multiplicative equation r^{-1}+Delta=-(v_2/v_1)(1+O(...)) would force r^{-1}=0.", "The final paragraph drops the beta in the assumed Delta bound and nevertheless concludes a bound with beta^2.", "The proof's conversion from a polynomial residual to geometric distance from the hyperbola requires a lower gradient estimate." ], "reasoning": "The formal zero-residual elimination does yield the displayed hyperbola. The tolerance estimates needed for the actual thickened incidence statement are not established." } }, { "id": "rmk:concentrated-surface", "kind": "remark", "label": "Remark 8.16", "title": "", "section": "sec-kakeya-model", "statement": "$\\quad$ \\item Note that $I(y_1, y_2)$ will change if we replace $y_1$ with $y'_1 \\in \\Omega(y_1)$. However, there exists a one-to-one map \\[ P_{y_1, y'_1}: I(y_1, y_2) \\to I(y'_1, y_2) \\] such that $|P_{y_1, y'_1} (r') - r'| \\le \\beta |r'|$ for every $r' \\in I(y_1,y_2)$. \\item If $|v_1/w_1| \\le e^{-s'}$ or $|w_1/v_1| \\le e^{-s'}$, then the neighborhood described in~\\eqref{eq:hyperbola} agrees with the $ \\beta^{-100} e^{-3s'}$ neighborhood of the straight line \\[ v_1^{-1}(X - X_0) + w_1^{-1}(Y - Y_0) = 0.\\]", "argument_gist": "$\\quad$ \\item Note that $I(y_1, y_2)$ will change if we replace $y_1$ with $y'_1 \\in \\Omega(y_1)$. However, there exists a one-to-one map \\[ P_{y_1, y'_1}: I(y_1, y_2) \\to I(y'_1, y_2) \\] such that $|P_{y_1, y'_1} (r') - r'| \\le \\beta |r'|$ for every $r' \\in I(y_1,y_2)$. \\item If $|v_1/w_1| \\le e^{-s'}$ or $|w_1/v_1| \\le e^{-s'}$, then the neighborhood described in~\\eqref{eq:hyperbola} agrees with the $ \\beta^{-100} e^{-3s'}$ neighborhood of the straight line \\[ v_1^{-1}(X - X_0) + w_1^{-1}(Y - Y_0) = 0.\\]", "gist_origin": "extractive_statement", "anchor": { "source": "manuscript", "line_start": 1934, "line_end": 1945, "char_start": 107930, "char_end": 108568 } }, { "id": "sec-dimension-improvement-unstructured", "kind": "section", "label": "Section 9", "title": "Dimension Improvement --- unstructured part", "section": "sec-dimension-improvement-unstructured", "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 1958, "line_end": 2378, "char_start": 109617, "char_end": 138682 } }, { "id": "prop:improve-dimension", "kind": "prop", "label": "Prop 9.1", "title": "", "section": "sec-dimension-improvement-unstructured", "statement": "\\par Suppose that $\\cF_i$ is $(d_i, s_i)$-good. Let $\\mu = a(s_i)_\\ast\\mu_{\\cF_i}$. Let us fix a full playground $\\Pi(x)$ and denote $\\tilde \\mu = \\mu_{\\Pi(x)}$. Then $\\tilde \\mu$ admits the following decomposition: \\[ \\tilde \\mu = \\nu_{\\cR_i} \\tilde \\mu_{\\cR_i} + \\nu_{i+1} \\tilde \\mu_{i+1}, \\] such that $\\nu_{\\cR_i} \\le \\beta^{20}$, and $\\mu_{i+1}$ is $(d_i+\\epsilon_1, s_{i+1})$-good. Recall that $s_{i+1} = s_i/2$.", "argument_gist": "Decomposes heavy neighborhoods into random, strange and structured classes and combines small-mass estimates for all three to improve transverse dimension.", "gist_origin": "review_model", "anchor": { "source": "manuscript", "line_start": 1962, "line_end": 1971, "char_start": 109775, "char_end": 110266 }, "review": { "verdict": "major_issues", "issues": [ "All three constituent estimates have unresolved prerequisites or proof gaps.", "The proposition omits the upper restriction on d_i required by prop:dimension-improvement-structured-component.", "The heavy-neighborhood argument does not justify the exact number of covering Theta packets or the asserted bound after renormalizing the complement.", "Definition def:structure-randomness assumes a packet bound without beta^{-15}, stronger than the preceding lm:omega-upper establishes." ], "reasoning": "The intended decomposition is a coherent strategy, but it is not a completed proof. The main theorem's intended stopping range can satisfy the structured proposition's restriction; that does not repair the unrestricted proposition as written." } }, { "id": "def:heavy-part", "kind": "defn", "label": "Defn 9.2", "title": "", "section": "sec-dimension-improvement-unstructured", "statement": "\\par For $z \\in \\cI$, we call $\\Theta'(z)$ a heavy neighborhood if \\[ \\tilde \\mu (\\Theta'(z)) > e^{-2s'} e^{- (d_i + \\epsilon_1) s'}. \\] $\\Theta'(z)$ a extremely heavy if \\[ \\tilde \\mu (\\Theta'(z)) > e^{-2s'} e^{- (d_i -C_1 \\epsilon_1) s'}. \\]", "argument_gist": "\\par For $z \\in \\cI$, we call $\\Theta'(z)$ a heavy neighborhood if \\[ \\tilde \\mu (\\Theta'(z)) > e^{-2s'} e^{- (d_i + \\epsilon_1) s'}. \\] $\\Theta'(z)$ a extremely heavy if \\[ \\tilde \\mu (\\Theta'(z)) > e^{-2s'} e^{- (d_i -C_1 \\epsilon_1) s'}. \\]", "gist_origin": "extractive_statement", "anchor": { "source": "manuscript", "line_start": 1978, "line_end": 1985, "char_start": 110484, "char_end": 110796 } }, { "id": "defn:line:1985", "kind": "defn", "label": "Defn 9.3", "title": "", "section": "sec-dimension-improvement-unstructured", "statement": "\\par Let $\\cR_i \\subset \\Pi(x)$ denote the union of heavy neighborhoods. Then $\\cR_i$ will be said to be the heavy part of $\\Pi(x)$.", "argument_gist": "\\par Let $\\cR_i \\subset \\Pi(x)$ denote the union of heavy neighborhoods. Then $\\cR_i$ will be said to be the heavy part of $\\Pi(x)$.", "gist_origin": "extractive_statement", "anchor": { "source": "manuscript", "line_start": 1985, "line_end": 1988, "char_start": 110797, "char_end": 110960 } }, { "id": "lm:omega-upper", "kind": "lemma", "label": "Lemma 9.4", "title": "", "section": "sec-dimension-improvement-unstructured", "statement": "Let us denote $s' = s_{i+1}$. For any $\\Omega(y) = \\Omega_{s'} y$, we have \\tilde \\mu (\\Omega'(y)) \\le \\beta^{-15}e^{-2 s'} e^{-2 d_i s'}.", "argument_gist": "Pulls Omega back to an isotropically thin transverse neighborhood and uses the short U-width and the previous dimension bound.", "gist_origin": "review_model", "proof_anchor": { "source": "manuscript", "line_start": 2006, "line_end": 2019, "char_start": 112148, "char_end": 112827 }, "proof_opening": "First note that \\[ \\mu(\\Omega(y)) = \\mu_{\\cF_i}(a(-2s') \\Omega(y)),\\] and \\[ a(-2s') \\Omega(y) = Q_{2s'}(2\\beta) R^{A_0 H}(2s', 2\\beta) y' \\] where $y' = a(-2s') y$. Noting that $\\mu_{\\cF_i}$ is a normalized measure on $U$-orbits, we have \\[ \\mu_{\\cF_i} (Q_{2s'}(2\\beta) R^{A_0 H}(2s', 2\\beta) y') \\le e^{-2s'} \\mu_{\\cF_i} (Q_{2s'}(2\\beta) B^{A_0 H }(2\\beta) y'). \\] By our hypothesis that $\\mu_{\\cF_i}$ is $(d_i, 2s')$-good, we have \\[ \\mu_{\\cF_i} (Q_{2s'}(2\\beta) B^{A_0 H }(2\\beta) y') \\le e^{-2d_i s'}, \\] which implies the claim. \\par Since $\\Pi(x)$ is a full playground, we have that for any $y \\in \\fJ$, \\eqref{eq:upper-omega} holds.", "anchor": { "source": "manuscript", "line_start": 1998, "line_end": 2006, "char_start": 111890, "char_end": 112147 }, "review": { "verdict": "major_issues", "issues": [ "The (d_i,2s')-good hypothesis is at radius beta, but the proof applies it directly at radius 2beta without a covering loss.", "The e^{-2s'} factor requires uniformity along appropriate complete U-pieces, which is not ensured after the exceptional removals.", "The later def:structure-randomness silently discards this lemma's beta^{-15} conditioning factor." ], "reasoning": "The conjugated transverse scales are correct. The measure estimate requires an orbitwise disintegration and radius-enlargement argument not supplied here." } }, { "id": "def:heavy-part-decomposition", "kind": "defn", "label": "Defn 9.5", "title": "", "section": "sec-dimension-improvement-unstructured", "statement": "\\par For integer $j \\ge 1$, let us denote \\[ \\Xi_j(z') := \\Xi_{s',j} z' \\cap \\Theta'(z'), \\] where \\[ \\Xi_{s', j} := Q^{s'}_{s' + j}Q^{A_0 H}(2s', 1.1\\beta).\\] \\par Given a heavy neighborhood $\\Theta' (z)$, we call $\\Theta'(z)$ strange if there exists $ \\Xi_{C_1 \\epsilon_1 s'}(z') \\subset \\Theta'(z)$ such that \\[ \\tilde \\mu_{\\fr}(\\Xi_{C_1 \\epsilon_1 s'}(z')) \\ge (1-\\beta) \\tilde \\mu_{\\fr}(\\Theta'(z)). \\] Otherwise, we call it a normal neighborhood. Let $\\bar \\cR_i \\subset \\cR_i$ denote the union of all the heavy neighborhoods that are also normal. Then we have the following decomposition: \\cR_i = \\bar \\cR_i \\cup \\bigcup_{j = C_1 \\epsilon_1 s'}^{2s'} \\cR_{i,j}, where $\\cR_{i,j}$ denotes the union of heavy neighborhoods $\\Theta'(z)$ such that there exists $\\Xi_j(z') \\subset \\Theta\u2019(z)$ satisfying \\tilde \\mu_{\\fr}(\\Xi_j(z')) \\ge (1-\\beta)^{j} \\tilde \\mu_{\\fr}(\\Theta'(z)), and \\tilde \\mu_{\\fr}(\\Xi_{j+1}(z'')) < (1-\\beta) \\tilde \\mu_{\\fr}(\\Xi_j(z')) for any $\\Xi_{j+1}(z'') \\subset \\Xi_j(z')$. Here $j \\le 2s'$ because for $j > 2s'$, \\[ \\Xi_j (z') \\subset \\Omega(z'), \\] and by \\eqref{eq:upper-omega}, \\[ \\tilde \\mu (\\Omega(z')) \\le \\beta^{-15}e^{-2 s'} e^{-2 d_i s'}, \\] which implies (since $\\Theta'(z)$ is heavy) that \\eqref{eq:strange-nbhd-condition} can not hold for $\\Xi_j(z')$ and~$\\Theta'(z)$.", "argument_gist": "\\par For integer $j \\ge 1$, let us denote \\[ \\Xi_j(z') := \\Xi_{s',j} z' \\cap \\Theta'(z'), \\] where \\[ \\Xi_{s', j} := Q^{s'}_{s' + j}Q^{A_0 H}(2s', 1.1\\beta).\\] \\par Given a heavy neighborhood $\\Theta' (z)$, we call $\\Theta'(z)$ strange if there exists $ \\Xi_{C_1 \\epsilon_1 s'}(z') \\subset \\Theta'(z)$ such that \\[ \\tilde \\mu_{\\fr}(\\Xi_{C_1 \\epsilon_1 s'}(z')) \\ge (1-\\beta) \\tilde \\mu_{\\fr}(\\Theta'(z)). \\] Otherwise, we call it a normal neighborhood. Let $\\bar \\cR_i \\subset \\cR_i$ denote the union of all the heavy neighborhoods that are also norm", "gist_origin": "extractive_statement", "anchor": { "source": "manuscript", "line_start": 2020, "line_end": 2047, "char_start": 112878, "char_end": 114493 } }, { "id": "def:structure-randomness", "kind": "defn", "label": "Defn 9.6", "title": "", "section": "sec-dimension-improvement-unstructured", "statement": "\\par Let $s'>0$, $\\Pi(x)$ and $\\tilde \\mu$ be as in Definition \\ref{def:kakeya-model}. To fit our case, let us assume that for any $y \\in \\fJ$, $$\\cW(y) = \\tilde \\mu(\\Omega'(y)) \\le e^{-2 d_i s' } e^{-2s'}.$$ By Lemma \\ref{lm:concentrated-surface}, for every $y \\in \\fJ$ and $z \\in \\cI$ from the same cross-section $\\Sigma^r$, there is a corresponding neighborhood $\\cC(\\cT(y), z) \\subset \\fC$. \\par Let us consider the summation \\[ \\fS(\\cT(y), z) := \\sum_{\\substack{ y' \\in \\cI_z \\\\ \\fD_z(y') \\in \\cC(\\cT(y), z)\\cap \\fB(\\cT(y), z) }} \\cW(y'). \\] Let $j$ be so that the size of the sides of $\\|(v_1, w_1)\\|$ are in $[e^{s'-j}, e^{s'-j+1}]$ (cf.~Lemma~\\ref{lm:concentrated-surface}, particularly equation \\eqref{eq:hyperbola-box}). Then it is easy to see that \\fS(\\cT(y), z) \\le N_2 \\beta^{-100} \\beta^{-100}e^{2s' - j} e^{-2 d_i s' } e^{-2s'} \\le \\beta^{-201} e^{-2 d_i s' - j}, where $N_2$ is a large numeric constant (in \\S\\ref{subsec-constants},it was chosen to be $10^6$). The curve neighborhood $\\cC(\\cT(y), z)$ is called highly concentrated if $$ \\fS(\\cT(y), z) \\ge e^{-2 d_i s' -7C_1\\epsilon_1 s'}.$$", "argument_gist": "\\par Let $s'>0$, $\\Pi(x)$ and $\\tilde \\mu$ be as in Definition \\ref{def:kakeya-model}. To fit our case, let us assume that for any $y \\in \\fJ$, $$\\cW(y) = \\tilde \\mu(\\Omega'(y)) \\le e^{-2 d_i s' } e^{-2s'}.$$ By Lemma \\ref{lm:concentrated-surface}, for every $y \\in \\fJ$ and $z \\in \\cI$ from the same cross-section $\\Sigma^r$, there is a corresponding neighborhood $\\cC(\\cT(y), z) \\subset \\fC$. \\par Let us consider the summation \\[ \\fS(\\cT(y), z) := \\sum_{\\substack{ y' \\in \\cI_z \\\\ \\fD_z(y') \\in \\cC(\\cT(y), z)\\cap \\fB(\\cT(y), z) }} \\cW(y'). \\] Let", "gist_origin": "extractive_statement", "anchor": { "source": "manuscript", "line_start": 2050, "line_end": 2067, "char_start": 114622, "char_end": 115932 } }, { "id": "remark:line:2069", "kind": "remark", "label": "Remark 9.7", "title": "", "section": "sec-dimension-improvement-unstructured", "statement": "Note that $\\fS(\\cT (y), z) $ is essentially the $\\tilde\\mu$-measure of a subset of $\\Theta'(z)$ with $r$ so that $z,y \\in \\Sigma^r$ (in particular, up to edge effects, $\\Theta'(z)$ is a subset of $\\Sigma^r$). Thus \\[ \\fS(\\cT(y),z) \\lessapprox \\tilde\\mu(\\Sigma^r) \\leq e^{-2s'}. \\] Thus, if $d_i<1$ there can be no highly concentrated curve neighborhoods. Moreover if $\\Theta'(z)$ is not extremely heavy, then unless $d_i >2-O(\\epsilon_1)$ then $ \\cC(\\cT(y),z)$ is not highly concentrated.", "argument_gist": "Note that $\\fS(\\cT (y), z) $ is essentially the $\\tilde\\mu$-measure of a subset of $\\Theta'(z)$ with $r$ so that $z,y \\in \\Sigma^r$ (in particular, up to edge effects, $\\Theta'(z)$ is a subset of $\\Sigma^r$). Thus \\[ \\fS(\\cT(y),z) \\lessapprox \\tilde\\mu(\\Sigma^r) \\leq e^{-2s'}. \\] Thus, if $d_i<1$ there can be no highly concentrated curve neighborhoods. Moreover if $\\Theta'(z)$ is not extremely heavy, then unless $d_i >2-O(\\epsilon_1)$ then $ \\cC(\\cT(y),z)$ is not highly concentrated.", "gist_origin": "extractive_statement", "anchor": { "source": "manuscript", "line_start": 2069, "line_end": 2076, "char_start": 115935, "char_end": 116472 } }, { "id": "def:structure-random-decompostion", "kind": "defn", "label": "Defn 9.8", "title": "", "section": "sec-dimension-improvement-unstructured", "statement": "\\par Given $z \\in \\cI$ such that $\\Theta'(z)$ is heavy and normal, and $\\Sigma^r$ such that $z \\in \\Sigma^r$, let us define: \\HS(z,r) := \\left\\{ y \\in \\fJ\\cap \\Sigma^r: \\cC(\\cT(y), z) \\subset \\fC \\text{ is highly concentrated} \\right\\}, and \\SS(z,r) := \\bigcup_{y \\in \\HS(z, r)} \\Omega'(y). If there exists $\\Sigma^r$ containing $z$ such that \\[ \\tilde \\mu (\\SS(z,r)) \\ge e^{-2s' - 10 C_1 \\epsilon_1 s'}, \\] we will call $\\Theta'(z)$ highly structured. Then we have the following decomposition for $\\bar \\cR_i$: \\bar \\cR_i = \\bar \\cR^{\\fr}_{i} \\cup \\bar \\cR^{\\fs}_{i}, where $\\bar \\cR^{\\fs}_i$ denotes the union of heavy, normal, and highly structured $\\Theta'(z)$, and $\\bar \\cR^{\\fr}_i$ denotes the union of heavy, normal $\\Theta'(z)$ which are not highly structured.", "argument_gist": "\\par Given $z \\in \\cI$ such that $\\Theta'(z)$ is heavy and normal, and $\\Sigma^r$ such that $z \\in \\Sigma^r$, let us define: \\HS(z,r) := \\left\\{ y \\in \\fJ\\cap \\Sigma^r: \\cC(\\cT(y), z) \\subset \\fC \\text{ is highly concentrated} \\right\\}, and \\SS(z,r) := \\bigcup_{y \\in \\HS(z, r)} \\Omega'(y). If there exists $\\Sigma^r$ containing $z$ such that \\[ \\tilde \\mu (\\SS(z,r)) \\ge e^{-2s' - 10 C_1 \\epsilon_1 s'}, \\] we will call $\\Theta'(z)$ highly structured. Then we have the following decomposition for $\\bar \\cR_i$: \\bar \\cR_i = \\bar \\cR^{\\fr}_{i} \\cup \\", "gist_origin": "extractive_statement", "anchor": { "source": "manuscript", "line_start": 2078, "line_end": 2098, "char_start": 116525, "char_end": 117611 } }, { "id": "prop:dimension-improvement-random", "kind": "prop", "label": "Prop 9.9", "title": "", "section": "sec-dimension-improvement-unstructured", "statement": "\\[ \\tilde \\mu (\\bar \\cR^{\\fr}_{i}) \\le \\beta^{22}. \\]", "argument_gist": "Obtains a lower weighted intersection count from recurrent visits to normal heavy neighborhoods, and an incompatible upper count from non-concentration near encounter hyperbolas.", "gist_origin": "review_model", "proof_anchor": { "source": "manuscript", "line_start": 2103, "line_end": 2219, "char_start": 117901, "char_end": 126851 }, "proof_opening": "\\par For a contradiction, let us assume $\\tilde \\mu(\\bar \\cR^{\\fr}_i) \\ge \\beta^{22}$. \\par For every $y \\in \\bar \\cR^{\\fr}_i$, if \\[ |u(\\fl(y))y \\cap \\bar \\cR^{\\fr}_i| \\ge \\beta^{31} e^{2s'}, \\] we will call it a good point. Otherwise, we call it a bad point. Then it is easy to see that the collection of bad points has $\\tilde \\mu$-measure $\\le \\beta^{30}$. Therefore, the collection of good points in $\\bar \\cR^{\\fr}_i$, denoted by $\\cR'_{i}$, has $\\tilde \\mu$-measure $\\ge \\beta^{22.1}$. \\par For $r \\in [\\beta e^{2s'}]$, we say that the cross-section $\\Sigma^r \\subset \\Pi(x)$ is a good cross-section if \\[ \\tilde \\mu_{\\fr}(\\cR'_i \\cap \\Sigma^r) \\ge \\beta^{30} \\tilde \\mu_{\\fr}(\\Sigma^r).\\] Oth", "anchor": { "source": "manuscript", "line_start": 2099, "line_end": 2103, "char_start": 117773, "char_end": 117900 }, "review": { "verdict": "major_issues", "issues": [ "The lower count relies on unproved packet mass transport, bounded overlap and residence-time estimates.", "The asserted beta-separated sample size is not guaranteed with the exact constants from the displayed length lower bound; a constant-factor loss is needed.", "The upper count uses the stronger packet-weight hypothesis introduced in def:structure-randomness, not the previously proved beta^{-15}-weaker estimate.", "The equality expressing the pair count as a sum over heavy neighborhood centers is not exact for overlapping covers; a bounded-multiplicity inequality must be proved.", "Several uses of mu_fr have no independently defined measure in this portion of the argument." ], "reasoning": "If the geometric and conditional-measure inputs were rigorously established with slack, the opposing exponents would express the intended contradiction. As supplied, those inputs and the counting identities are not justified." } }, { "id": "prop:dimension-improvement-strange-part", "kind": "prop", "label": "Prop 9.10", "title": "", "section": "sec-dimension-improvement-unstructured", "statement": "For any $ C_1 \\epsilon_1 s' \\le j \\le 2s' $, we have \\[ \\tilde{\\mu}(\\cR_{i,j}) \\le \\beta^{22}. \\]", "argument_gist": "Repeats the weighted incidence argument for packets concentrated at an intermediate contracting scale j, gaining an extra initial-dimension factor e^{-d_0 j}.", "gist_origin": "review_model", "proof_anchor": { "source": "manuscript", "line_start": 2225, "line_end": 2372, "char_start": 127184, "char_end": 138111 }, "proof_opening": "For each $\\Theta'(\\bar z) \\subset \\cR_{i,j}$, fix a subset $\\Xi_j(z) \\subset \\Theta'(\\bar z)$ as in \\eqref{eq:strange-nbhd-condition} and let $\\cR'_{i,j}$ denote the union of all of those $\\Xi_j(z)$ for $\\Theta'(\\bar z) \\subset \\cR_{i,j}$. To prove $\\tilde \\mu_{\\fr}(\\cR_{i,j}) < \\beta^{22} $, it suffices to show that $\\tilde \\mu_{\\fr}(\\cR'_{i,j}) < \\beta^{22.1}$. \\par Assume in contradiction that $\\tilde \\mu_{\\fr}(\\cR'_{i,j}) \\ge \\beta^{22.1}$. Using similar arguments as in the proof of Proposition \\ref{prop:dimension-improvement-random}, we can get a subset $\\cR''_{i,j} \\subset \\cR'_{i,j} $ such that $\\tilde \\mu_{\\fr}(\\cR''_{i,j}) \\ge \\beta^{22.2}$ and for any $y \\in \\cR''_{i,j}$, \\[|u(\\fl(", "anchor": { "source": "manuscript", "line_start": 2220, "line_end": 2225, "char_start": 127001, "char_end": 127183 }, "review": { "verdict": "major_issues", "issues": [ "The proof initially establishes a visit length beta^{30}e^{2s'} but then asserts the stronger beta^{22}e^{2s'} lower bound. For beta<1 this does not follow.", "The precise annular Delta condition is not proved from membership in Xi_j minus Xi_{j+1}; changes of packet representatives and time changes introduce constants.", "The localization lemma and the initial translated dimension estimate used in the upper bound are not established.", "The decomposition into strange scales is described by nested Xi sets without proving that an appropriate nested selection exists in these non-partition neighborhoods.", "The incidence and overlap issues from the random-part proof remain." ], "reasoning": "The terminal comparison using d_0 C_1=10 is consistent. It cannot certify the preceding lower and upper estimates." } }, { "id": "lem:sigma j lemma", "kind": "lemma", "label": "Lemma 9.11", "title": "", "section": "sec-dimension-improvement-unstructured", "statement": "If there is at least one $y_1\\in \\cI_{ z} \\cap \\Xi_j (z')$ so that $(y_2, y_1) \\in \\diamond_{r,j}$ then \\[ z' \\in \\Sigma_{s', j-1} (y_2). \\]", "argument_gist": "Combines small relative contracting slope and late encounter geometry to localize an entire heavy slice in a smaller translated transverse neighborhood.", "gist_origin": "review_model", "proof_anchor": { "source": "manuscript", "line_start": 2298, "line_end": 2328, "char_start": 132926, "char_end": 135190 }, "proof_opening": "Since $z', y_2 \\in \\Sigma^r$, it suffices to show that \\[ z' \\in \\Pi_{s', j-1}(y_2). \\] Write \\[ y_1 = a_0 a u^\\ast (r^\\ast_{y_1})\\exp(\\vv w^-) \\exp(\\vv w^+) u(r_{y_1}) y_2, \\] where $\\vv w^- \\in \\vv r^-$ satisfies that $\\|\\vv w^-\\| \\in [\\beta e^{-s' - j-1}, \\beta e^{-s' - j }]$. Then by Lemma~\\ref{lm:concentrated-surface}, we have $\\|\\vv w^+\\| \\le \\beta e^{s' - j}$. Let $\\hat y_1 = a(-2t) y_1, \\hat y_2 = a(-2t) y_2$. Then we have \\[ \\hat y_1 = a_0 a u^\\ast (\\hat r^\\ast) \\exp (\\hat{ \\vv w}^-)\\exp (\\hat{ \\vv w}^+) u(\\hat r) \\hat y_2,\\] where $\\hat r^\\ast, \\hat r \\in [\\beta]$, $\\hat{\\vv w}^- \\in B_{\\vv r^-}(\\beta e^{-j})$, and $\\hat{\\vv w}^+ \\in B_{\\vv r^+}(\\beta e^{-j})$. By Lemma \\ref{lm:pro", "anchor": { "source": "manuscript", "line_start": 2293, "line_end": 2298, "char_start": 132614, "char_end": 132925 }, "review": { "verdict": "major_issues", "issues": [ "The proof defines hat y_i using a(-2t), but all following estimates require a(-2s'). These conjugations have different transverse scales.", "The final relative block-diagonal radius is 2.1beta, whereas Pi_{s',j-1} still has radius beta in those directions. Decreasing j enlarges only the transverse radii and does not absorb this discrepancy.", "The contracting-coordinate estimates depend on the unresolved concentrated-surface lemma." ], "reasoning": "Replacing t by s' addresses one evident typographical error, but the remaining neighborhood containment still does not follow from the displayed bounds." } }, { "id": "sec-no-loss-of-dimension", "kind": "section", "label": "Section 10", "title": "No loss of dimension in a coarser scale", "section": "sec-no-loss-of-dimension", "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 2378, "line_end": 2553, "char_start": 138682, "char_end": 149994 } }, { "id": "prop:dimension-does-not-drop", "kind": "prop", "label": "Prop 10.1", "title": "", "section": "sec-no-loss-of-dimension", "statement": "Let us keep the notation as in Proposition \\ref{prop:improve-dimension}. Then there exists $\\mathfrak{Z} \\subset \\Pi(x)$ with $\\tilde \\mu(\\mathfrak Z) \\le \\beta^{50}$ such that for $x \\in \\cI\\setminus\\mathfrak{Z}$, \\tilde \\mu(\\Theta'(x)) \\le e^{-2s'} e^{-(d_i - C_1 \\epsilon_1)s'}.", "argument_gist": "Repeats the normal and strange weighted-intersection estimates at a higher heaviness threshold to show that extremely heavy neighborhoods have small total mass.", "gist_origin": "review_model", "proof_anchor": { "source": "manuscript", "line_start": 2389, "line_end": 2540, "char_start": 139446, "char_end": 149415 }, "proof_opening": "\\par For $z \\in \\cI$, let us call $\\Theta'(z)$ extremely heavy if \\[ \\tilde \\mu(\\Theta'(z)) > \\tilde M_{s'} \\qquad\\text{for $ \\tilde M_{s'}= e^{-2s'} e^{-(d_i - C_1 \\epsilon_1)s'}$}. \\] Let $\\mathfrak Z$ denote the union of extremely heavy $\\Theta'(z)$'s. Then it suffices to show that \\[ \\tilde \\mu (\\mathfrak Z ) \\le \\beta^{50}. \\] Similarly to \\eqref{eq:strange-normal-decomposition}, let us decompose \\[ \\fZ = \\fZ' \\cup \\bigcup_{ C_1 \\epsilon_1 s' \\le j \\le 2s'} \\fZ_j. \\] Here $\\fZ'$ denotes the union of extremely heavy $\\Theta'(z)$ such that for any $$\\Xi_{C_1 \\epsilon_1 s'} (y) \\subset \\Theta'(z),$$ we have \\[ \\tilde \\mu(\\Xi_{C_1 \\epsilon_1 s'} (y)) \\le (1- \\beta) \\tilde \\mu (\\Theta'(z)); ", "anchor": { "source": "manuscript", "line_start": 2381, "line_end": 2389, "char_start": 139038, "char_end": 139445 }, "review": { "verdict": "major_issues", "issues": [ "This is only a sketch and inherits the unresolved geometry and conditioning estimates of both preceding incidence proofs.", "The definition of diamond'_{r,j} tests intersection with Z' rather than the strange-scale set Z'_j used to construct the lower bound.", "The count of heavy slices again requires bounded overlap rather than disjointness, and normalization factors are inconsistently omitted.", "Approximate comparison notation is used without quantified losses in estimates essential to the contradiction." ], "reasoning": "The higher-threshold strategy is intelligible and there is no apparent circular dependency: it is intended to precede the structured-part proof. Its proof is nevertheless incomplete." } }, { "id": "sec-structured-component", "kind": "section", "label": "Section 11", "title": "Dimension Improvement --- structured part", "section": "sec-structured-component", "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 2553, "line_end": 3240, "char_start": 149994, "char_end": 196082 } }, { "id": "prop:dimension-improvement-structured-component", "kind": "prop", "label": "Prop 11.1", "title": "", "section": "sec-structured-component", "statement": "Suppose $ d_i \\le 4 - 10^{11} C_1 \\epsilon_1 $. We have \\[ \\tilde \\mu (\\bar \\cR^{\\fs}_i) \\le \\beta^{22}. \\]", "argument_gist": "Turns a large family of concentrated encounter curves into many occupied fine packets, unless the curves lie in narrow strips corresponding to SL(2,R) semidirect R^2 neighborhoods; rules out those strips by subgroup non-concentration.", "gist_origin": "review_model", "proof_anchor": { "source": "manuscript", "line_start": 2937, "line_end": 3225, "char_start": 174073, "char_end": 195502 }, "proof_opening": "\\par Assume in contradiction that \\[ \\tilde \\mu (\\bar \\cR^{\\fs}_i) > \\beta^{22}. \\] Let $\\fZ \\subset \\Pi(x)$ be the subset from Proposition \\ref{prop:dimension-does-not-drop}. By replacing $\\bar \\cR^{\\fs}_i$ with $\\bar \\cR^{\\fs}_i \\setminus \\mathfrak{Z}$, we can assume that \\eqref{eq:upper-theta} holds for any $z \\in \\cI \\cap \\bar\\cR^{\\fs}_i$. Thus we may assume: \\cI \\cap \\bar \\cR^{\\fs}_i \\neq \\emptyset \\\\ \\tilde \\mu ( \\Theta'(z)) \\le e^{-2s'} e^{-(d_i - C_1 \\epsilon_1) s'} \\qquad \\text{for any $z \\in \\cI \\cap \\bar \\cR^{\\fs}_i$} { \\par Let us fix a $z \\in \\cI\\cap \\bar \\cR^{\\fs}_i$. By the definition of $\\bar \\cR^{\\fs}_i$ (see Definition \\ref{def:structure-random-decompostion}), we have that ", "anchor": { "source": "manuscript", "line_start": 2562, "line_end": 2567, "char_start": 150289, "char_end": 150486 }, "review": { "verdict": "major_issues", "issues": [ "The curve-comparison package depends on the false implication in lm:hyperbola-plane and the unproved counting lemmas.", "The subgroup non-concentration input has a reversed conjugation in its proof.", "The asserted passage from a concentrated curve to at least e^{2s'-9C_1 epsilon_1 s'} full packets needs a separate estimate on almost-empty packets and overlap; it is not demonstrated.", "Curves passing within a neighborhood of a grid point are subsequently treated as exact hyperbolas with that common center. The parameter errors from this recentering are not controlled.", "Removing curves contained in one narrow strip does not imply that the next strip obtained is disjoint from the first. The repeated-strip construction and its count of distinct full packets require an additional selection argument.", "The greedy construction guarantees new points relative to earlier selected curves, not points uncovered by every other selected curve as later asserted. A disjoint charging scheme would have to be explicitly maintained." ], "reasoning": "This is the novel bottleneck of the proposed argument. The large exponent margins may accommodate some local losses, but they do not replace the missing geometric selection and counting arguments." } }, { "id": "def:full-omega", "kind": "defn", "label": "Defn 11.2", "title": "", "section": "sec-structured-component", "statement": "For $z \\in \\fJ$, if \\tilde \\mu(\\Omega'(z)) \\ge e^{-2s'} e^{-2d_i s' - 8 C_1 \\epsilon_1 s'}, we will call $\\Omega'(z)$ full. Otherwise, we call it almost empty.", "argument_gist": "For $z \\in \\fJ$, if \\tilde \\mu(\\Omega'(z)) \\ge e^{-2s'} e^{-2d_i s' - 8 C_1 \\epsilon_1 s'}, we will call $\\Omega'(z)$ full. Otherwise, we call it almost empty.", "gist_origin": "extractive_statement", "anchor": { "source": "manuscript", "line_start": 2572, "line_end": 2580, "char_start": 151254, "char_end": 151533 } }, { "id": "def:comparable", "kind": "defn", "label": "Defn 11.3", "title": "", "section": "sec-structured-component", "statement": "For $z \\in \\cI$, we say that two curve neighborhoods $\\cC = \\cC(\\cT(y), z)$ and $\\cC' = \\cC(\\cT(y'), z)$ are comparable if $\\cC'$ is contained in a $e^{-3s' + 11 C_1 \\epsilon_1 s'}$-neighborhood of $\\cC$. Let us denote by \\[ \\cC' \\approx \\cC, \\] if $\\cC$ and $\\cC'$ are comparable.", "argument_gist": "For $z \\in \\cI$, we say that two curve neighborhoods $\\cC = \\cC(\\cT(y), z)$ and $\\cC' = \\cC(\\cT(y'), z)$ are comparable if $\\cC'$ is contained in a $e^{-3s' + 11 C_1 \\epsilon_1 s'}$-neighborhood of $\\cC$. Let us denote by \\[ \\cC' \\approx \\cC, \\] if $\\cC$ and $\\cC'$ are comparable.", "gist_origin": "extractive_statement", "anchor": { "source": "manuscript", "line_start": 2580, "line_end": 2586, "char_start": 151534, "char_end": 151879 } }, { "id": "lm:hyperbola-plane", "kind": "lemma", "label": "Lemma 11.4", "title": "", "section": "sec-structured-component", "statement": "Let $C \\subset \\fC = \\left[1.2 \\beta e^{-s'}\\right]^2$ be a hyperbola with the following equation: \\[ X = \\frac{-Y}{w_1 Y + k}, \\] where $|w_1| \\in [e^{s' - 10 C_1 \\epsilon_1 s'}, \\beta e^{s'}]$, and $|k| \\in [e^{\\ell}, e^{\\ell+1}]$ for some $\\ell \\in \\Z_{\\ge 0}, \\ell \\le s'$. Let $C' \\subset \\fC$ be another hyperbola with the following equation: \\[ X = \\frac{-Y}{w'_1 Y + k'},\\] where $|w'_1| \\in [e^{s' - 10 C_1 \\epsilon_1 s'}, \\beta e^{s'}]$. \\par (1) For $0< A < e^{2s'-C_1\\epsilon_1 s'}$, if $C'$ is contained in the $A e^{-3s'}$ neighborhood of $C$, then we have |w_{1} - w'_{1}| \\le Ae^{-s'+2\\ell}, \\text{ and } |k - k'| \\le Ae^{-2s' +2\\ell}. Conversely, if \\eqref{eq:w1w'1} holds, then $C'$ is contained in the $A e^{-3s'}$ neighborhood of $C$. \\par (2) Suppose $C'$ is not contained in the $e^{-3s' + 11 C_1 \\epsilon_1 s'}$ neighborhood of $C$. For $0< A < e^{2s'-C_1\\epsilon_1 s'}$, if we have |w_{1} - w'_{1}| \\in \\left[Ae^{-s'+2\\ell} , Ae^{-s'+2\\ell+1}\\right], and there exists $ \\vv p' \\in C'$ such that $\\|\\vv p'\\| \\ge e^{-s' -10 C_1 \\epsilon_1 s'}$ and $\\vv p'$ is contained in the $\\beta^{-100} e^{-3s'}$ neighborhood of $C$, then $C'$ is contained in the $\\beta^{-100}A e^{-3s'}$ neighborhood of $C$. Moreover, if $\\vv p'_1, \\vv p'_2\\in C'$ satisfy that $\\|\\vv p'_1\\| , \\|\\vv p'_2\\| \\ge e^{-s' -10 C_1 \\epsilon_1 s'}$ and $\\vv p'_1, \\vv p'_2$ are both contained in the $\\beta^{-100} e^{-3s'}$ neighborhood of $C$, then $$\\|\\vv p'_1 - \\vv p'_2\\| \\le A^{-1} e^{-s' + 11 C_1 \\epsilon_1 s'}.$$", "argument_gist": "Relates closeness and intersection lengths of rational graphs X=-Y/(wY+k) to closeness of their parameters.", "gist_origin": "review_model", "anchor": { "source": "manuscript", "line_start": 2587, "line_end": 2607, "char_start": 151914, "char_end": 153679 }, "review": { "verdict": "major_issues", "issues": [ "Part (1)'s necessary parameter bounds omit factors depending on beta and are false with the stated constants. Set ell=0, k=k'=1, w=beta e^{s'}/2 and w'=w+2e^{-s'}, with admissible sufficiently large s', and A=1. Across the box, graph displacement is O(beta^2 e^{-3s'}), hence less than e^{-3s'} for small beta, but |w-w'|=2e^{-s'} exceeds the asserted e^{-s'} bound.", "The proof treats information over a Y-interval of size beta e^{-s'} as though its size were e^{-s'}, precisely losing beta-dependent factors in the necessary estimates.", "Euclidean neighborhood inclusion is replaced by vertical graph comparison without proving a uniform equivalence.", "Several algebraic estimates divide by w-w' without separately treating equality.", "Part (2)'s reduction from a lower bound on the point norm to a lower bound on |Y| requires slope bounds for the second graph that are not established." ], "reasoning": "The counterexample uses graphs of uniformly bounded slope and denominators near 1, so the small graph-displacement bound also gives the required geometric closeness after clipping to the square. This disproves the literal quantitative implication, although a weakened estimate with beta losses could be investigated." } }, { "id": "lm:integrable-surface", "kind": "lemma", "label": "Lemma 11.5", "title": "", "section": "sec-structured-component", "statement": "For $y \\in \\fJ\\cap \\Sigma^r , z \\in \\cI\\cap\\Sigma^r$, let us denote \\[\\cC = \\cC(\\cT(y), z)\\subset \\fC = \\left[1.2 \\beta e^{-s'}\\right]^2.\\] Suppose that $\\cC$ has equation \\[ (X- X_0) (Y- Y_0) + v_1^{-1} (X- X_0) + w_1^{-1} (Y- Y_0) = 0, \\] with $\\| (v_1 , w_1) \\| \\ge e^{s' - 10 C_1 \\epsilon_1 s'}$, and $$\\max\\{|v_1/w_1|, |w_1/v_1|\\} \\in [e^\\ell, e^{\\ell + 1}].$$ Then for $\\ell < s'$, there exist at most $e^{2s' + 2\\ell + 60C_1\\epsilon_1 s'}$ different $y' \\in \\fJ\\cap\\Sigma^r$ such that \\[\\cC(\\cT(y'), z) \\approx \\cC.\\] \\par For $\\ell \\ge s' $, there are at most $e^{4s' + 60 C_1 \\epsilon_1 s'}$ different $y' \\in \\fJ \\cap \\Sigma^r$ such that \\[ \\cC(\\cT(y'), z) \\approx \\cC. \\]", "argument_gist": "Counts packet labels inducing comparable encounter curves by fixing an approximate center, restricting rational-graph parameters, and counting admissible grid configurations.", "gist_origin": "review_model", "proof_anchor": { "source": "manuscript", "line_start": 2719, "line_end": 2775, "char_start": 160216, "char_end": 164172 }, "proof_opening": "\\par We first consider the case $\\ell < s'$. For $A > 1$, let us denote by $\\cC[A]$ the $Ae^{-3s'}$ neighborhood of $\\cC$. \\par Let us fix $z_0 \\in \\cI_z$ such that $\\fD_z(z_0) = (X'_0, Y'_0) \\in \\cC[e^{11 C_1 \\epsilon_1 s'}]$, and consider all $y' \\in \\fJ\\cap \\Sigma^r$ such that \\[ \\cC \\approx \\cC(\\cT(y'), z). \\] Assume in addition that $y'$ is such that there are coefficients $v_{1, y'},w_{1, y'}$ of size $\\leq \\beta e^{s'}$ and a point $y'_0 \\in \\Omega(y')$ such that \\[ z_0 = \\exp(v_{1, y'} \\vv v_1 + w_{1, y'} \\vv w_1) u(r_{y'}) y'_0. \\] Let us denote by $\\cM(\\cC, z_0)$ the collection of those $y'$'s. \\par Fix a $y \\in \\cM(\\cC,z_0)$ and consider the curve neighborhood $\\cC'$ with equation", "anchor": { "source": "manuscript", "line_start": 2704, "line_end": 2717, "char_start": 159423, "char_end": 160213 }, "review": { "verdict": "major_issues", "issues": [ "Uses the false necessary parameter estimate from lm:hyperbola-plane(1).", "The counts of y' in M(C,z_0) are asserted without deriving the relationship between the original packet grid and the parameters w,k,r.", "It is not proved that all comparable curves can be represented through one of the selected z_0 with the stipulated exact relation.", "There is an unexplained improvement from an 11C_1 epsilon_1 tolerance to a 10C_1 epsilon_1 tolerance in the proof." ], "reasoning": "The proposed parameter-counting mechanism is not a substitute for the omitted discretization estimates. The numerical bounds are unverified." } }, { "id": "lm:determine-curve-nbhd", "kind": "lemma", "label": "Lemma 11.6", "title": "", "section": "sec-structured-component", "statement": "\\par Let us fix constants $\\ell, m \\ge 1$. Let $\\cC \\subset \\fC$ be the $\\beta^{-100} e^{-3s'}$-neighborhood of the curve \\[ (X- X_0) (Y- Y_0) + v_1^{-1} (X- X_0) + w_1^{-1} (Y- Y_0) = 0, \\] with $\\|(v_1, w_1)\\| \\ge e^{s' - 10C_1 \\epsilon_1 s'}$ and \\[ \\max\\{|v_1/w_1|, |w_1/v_1|\\} \\in [e^\\ell, e^{\\ell + 1}].\\] Let $\\cC'$ be the $\\beta^{-100} e^{-3s'}$-neighborhood of the curve with the following equation \\[(X- X_0) (Y- Y_0) + {v'_1}^{-1}(X- X_0) + {w'_1}^{-1}(Y- Y_0) = 0.\\] Suppose we have that $\\cC$ and $\\cC'$ are not comparable. Moreover, suppose we have the following: \\[ \\max\\{|v'_1/w'_1|, |w'_1/v'_1|\\} \\in [e^\\ell, e^{\\ell + 1}],\\] \\[ \\max\\{ |v'_1 - v_1|, |w'_1 - w_1| \\} \\in [e^{s'-m}, e^{s'-m+1}], \\] and that there exists $(X_1, Y_1) \\in \\cC \\cap \\cC'$ with \\[ \\max\\{ |X_0 - X_1|, |Y_0 - Y_1| \\} \\ge e^{-s' - 10 C_1 \\epsilon_1 s'}. \\] Then $m + 2 \\ell \\le 2s' - 10 C_1 \\epsilon_1 s'$, and $\\cC'$ is contained in the $e^{-s'-m-2\\ell }$ neighborhood of $\\cC$.", "argument_gist": "Uses a noncentral intersection and a parameter-difference scale to place one encounter curve in a controlled neighborhood of another.", "gist_origin": "review_model", "proof_anchor": { "source": "manuscript", "line_start": 2793, "line_end": 2805, "char_start": 165224, "char_end": 166217 }, "proof_opening": "\\par Without loss of generality, we can assume that $(X_0, Y_0) = \\vv 0$ and $|w_1/v_1| \\in [e^\\ell, e^{\\ell+1}]$. Then we have $|w'_1/v'_1| \\in [e^\\ell, e^{\\ell+1}]$, $|w_1 - w'_1| \\in [e^{s'-m}, e^{s'-m+1}]$, and there exists $(X_1, Y_1) \\in \\cC \\cap \\cC'$ with $|Y_1| \\ge e^{-s'-10C_1 \\epsilon_1 s'}$. \\par Let us rewrite the equation of $\\cC$ as \\[ X = \\frac{-Y}{w_1 Y + k}, \\text{ where } k = w_1/ v_1, \\] and the equation of $\\cC'$ as \\[ X = \\frac{-Y}{w'_1 Y + k'}, \\text{ where } k' = w'_1/ v'_1. \\] Since $(X_1, Y_1) \\in \\cC \\cap \\cC'$, we have \\[ \\left|\\frac{Y_1}{w_1 Y_1 + k} - \\frac{Y_1}{w'_{1}Y_1 + k'} \\right| \\le \\beta^{-100}e^{-3s'}. \\] By Lemma \\ref{lm:hyperbola-plane}(2) with $A = e", "anchor": { "source": "manuscript", "line_start": 2777, "line_end": 2793, "char_start": 164175, "char_end": 165223 }, "review": { "verdict": "major_issues", "issues": [ "The cited part (2) yields a beta^{-100}A e^{-3s'} neighborhood, not the stronger A e^{-3s'} neighborhood claimed here.", "A maximum bound on changes in v and w is replaced without proof by the same scale bound on w alone.", "The hypotheses needed to apply lm:hyperbola-plane to the primed parameters are not all stated.", "The proof depends on the unresolved graph-comparison lemma." ], "reasoning": "Even accepting the cited lemma provisionally, the displayed conclusion drops an explicit factor beta^{-100}. Thus the asserted inference is not valid as written." } }, { "id": "lm:counting-curves-nbhd", "kind": "lemma", "label": "Lemma 11.7", "title": "", "section": "sec-structured-component", "statement": "\\par Let us fix constants $m, \\ell \\ge 1$ with $m+2\\ell \\le 2s' - 10 C_1 \\epsilon_1 s'$. Let $\\cC \\subset \\fC$ be the $\\beta^{-100} e^{-3s'}$-neighborhood of the curve \\[ (X- X_0) (Y- Y_0) + v_1^{-1} (X- X_0) + w_1^{-1} (Y- Y_0) = 0, \\] with $\\|(v_1, w_1)\\| \\ge e^{s' - 10C_1 \\epsilon_1 s'}$ and \\[ \\max\\{|v_1/w_1|, |w_1/v_1|\\} \\in [e^\\ell, e^{\\ell + 1}].\\] Let $\\fN( \\cC, (X_0, Y_0), m)$ denote a collection of non-comparable $\\beta^{-100} e^{-3s'}$-curve neighborhoods with equations of the form \\[(X- X_0) (Y- Y_0) + {v'_1}^{-1} (X- X_0) + {w'_1}^{-1}(Y- Y_0) = 0,\\] and contained in the $e^{-s'-m-2\\ell}$ neighborhood of $\\cC$. Then \\[ |\\fN( \\cC, (X_0, Y_0), m) | \\le e^{4s' - 2m - 4 \\ell + 100 C_1 \\epsilon_1 s'}.\\] Moreover, for $\\cC' \\in \\fN( \\cC, (X_0, Y_0), m)$ with corresponding parameters $v'_1,w'_1$, let $m'$ denote the number satisfying \\[ \\max \\{|w'_1 - w_1|, |v'_1 - v_1| \\} \\in [e^{s'-m'}, e^{s'-m'+1}]. \\] Then the intersection $$\\cC\\cap \\cC' \\cap \\left\\{ (X, Y): \\|(X, Y) - (X_0, Y_0)\\| \\ge e^{-s' - 10 C_1 \\epsilon_1 s'} \\right\\}$$ contains at most $e^{m'+ 2\\ell + 11 C_1 \\epsilon_1 s'}$ different points from the set \\[ \\left\\{ \\fD_z(z') : z' \\in \\cI_z \\right\\}. \\]", "argument_gist": "Packs admissible rational-graph parameters at the comparability scale and uses a short-intersection estimate to bound occupied grid points.", "gist_origin": "review_model", "proof_anchor": { "source": "manuscript", "line_start": 2824, "line_end": 2860, "char_start": 167471, "char_end": 169370 }, "proof_opening": "\\par Without loss of generality, we can assume that $(X_0, Y_0)= \\vv 0$, and $|w_1| \\ge |v_1|$. \\par Let us write the equation of $\\cC$ as \\[ X = \\frac{-Y}{w_1 Y + k}, \\text{ where } k = w_1/ v_1. \\] Then $|k| \\in [e^\\ell, e^{\\ell +1}]$. Suppose $\\cC'$ with equation \\[ X = \\frac{-Y}{w'_1 Y + k'} \\text{ where } k' = w'_1/v'_1 \\] is in $\\fN(\\cC, \\vv 0, m)$. Then we have for all $Y \\in [ \\beta e^{-s'}]$, \\[ \\left|\\frac{Y}{w_1 Y + k} - \\frac{Y}{w'_{1}Y + k'} \\right| \\le e^{-s' - m - 2\\ell}.\\] By Lemma \\ref{lm:hyperbola-plane} (1) with $A = e^{2s' - m - 2 \\ell}$, we have |w_1 - w'_{1}| \\le e^{s' -m }, \\text{ and } |k - k'| \\le e^{-m }. Note that by Lemma \\ref{lm:hyperbola-plane} (1) with $A = e^{", "anchor": { "source": "manuscript", "line_start": 2805, "line_end": 2824, "char_start": 166218, "char_end": 167470 }, "review": { "verdict": "major_issues", "issues": [ "The parameter packing relies on the false necessity direction of lm:hyperbola-plane(1).", "The collection's stated hypotheses do not impose all the parameter-range assumptions later used for its members.", "The final number of grid points is inferred from an intersection neighborhood without a complete grid-density estimate, including the beta-dependent mesh and thickening.", "Replacing a maximum parameter difference by the w-difference needed in part (2) is again unproved." ], "reasoning": "A packing argument could in principle prove a related bound with appropriate parameter restrictions and losses. This proof does not establish the stated one." } }, { "id": "lm:curve-in-N", "kind": "lemma", "label": "Lemma 11.8", "title": "", "section": "sec-structured-component", "statement": "Given $(X_0, Y_0) \\in \\fC = \\left[1.2\\beta e^{-s'}\\right]^2$, and $1 < \\ell \\le s'$, let us denote \\[ \\cN^\\ell = \\cN^{\\ell}(X_0, Y_0) := (X_0, Y_0) + [\\beta e^{-s' - \\ell}] \\times [\\beta e^{-s'}]. \\] Suppose we have a cross-section $\\Sigma^r$ and $z \\in \\cI \\cap \\Sigma^r$, $y \\in \\fJ\\cap\\Sigma^r$ such that $$\\cC = \\cC(\\cT(y),z) \\subset \\cN^{\\ell}.$$ Then there exists $z' \\in \\Theta(z)$ only depending on $\\cN^\\ell$ (and not on $\\cC$) such that $\\fD_z(z') \\in \\cN^{\\ell}$ and \\[ \\Omega'(y) \\subset a(2s')Q(\\fr_1, \\ell-1, \\beta ) B^{A_0 H}(\\beta) a(-2s') z'. \\]", "argument_gist": "Converts confinement of an encounter curve to a narrow vertical strip into small fr_1 coordinates, identifying concentration near a translated SL(2,R) semidirect R^2 piece.", "gist_origin": "review_model", "anchor": { "source": "manuscript", "line_start": 2861, "line_end": 2871, "char_start": 169405, "char_end": 170037 }, "review": { "verdict": "major_issues", "issues": [ "The asserted existence of Y with |Y-Y_1|>=beta e^{-s'} is not guaranteed on a centered interval of length beta e^{-s'}; a constant loss is necessary.", "The inference |k|>=e^ell from |w(Y-Y_1)+k|>=e^ell and |w(Y-Y_1)|<=beta^2 loses a subtraction.", "The BCH step is written for unscaled vectors but then uses bounds established only for their conjugated, beta-sized versions.", "The passage from one selected tilde y to the entire Omega'(y) needs a quantitative anisotropic product estimate, especially when ell approaches s'.", "The proof inherits the unproved encounter-curve representation." ], "reasoning": "The subgroup interpretation is a useful structural observation. The literal containment and constants do not follow from the supplied calculations." } }, { "id": "rmk:sl2r2", "kind": "remark", "label": "Remark 11.9", "title": "", "section": "sec-structured-component", "statement": "Lemma \\ref{lm:curve-in-N} roughly says that given $z \\in \\cI\\cap \\Sigma^r$ and $$ \\cN^\\ell = \\cN^\\ell(X_0, Y_0) \\subset \\fC$$ as above, all $y \\in \\fJ\\cap \\Sigma^r$ satisfying that \\[ \\cC(\\cT(y), z) \\subset \\cN^\\ell\\] is contained in a $a(2s')$-translate of a $\\beta e^{-\\ell+1}$-neighborhood of a piece of some $\\SL(2,\\R)\\ltimes \\R^2$-orbit of size $\\beta$.", "argument_gist": "Lemma \\ref{lm:curve-in-N} roughly says that given $z \\in \\cI\\cap \\Sigma^r$ and $$ \\cN^\\ell = \\cN^\\ell(X_0, Y_0) \\subset \\fC$$ as above, all $y \\in \\fJ\\cap \\Sigma^r$ satisfying that \\[ \\cC(\\cT(y), z) \\subset \\cN^\\ell\\] is contained in a $a(2s')$-translate of a $\\beta e^{-\\ell+1}$-neighborhood of a piece of some $\\SL(2,\\R)\\ltimes \\R^2$-orbit of size $\\beta$.", "gist_origin": "extractive_statement", "anchor": { "source": "manuscript", "line_start": 2924, "line_end": 2933, "char_start": 173554, "char_end": 173967 } }, { "id": "sec-main-proof", "kind": "section", "label": "Section 12", "title": "Proof of the main theorem", "section": "sec-main-proof", "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3240, "line_end": 3562, "char_start": 196082, "char_end": 211793 } }, { "id": "prop:high-dim-decomposition", "kind": "prop", "label": "Prop 12.1", "title": "", "section": "sec-main-proof", "statement": "\\par Let $s \\le \\hat s < t$, $s' =(t-\\hat s)/2$, and $\\hat \\mu$ be a probability measure on $X_{\\beta^{1/4}}$ defined by the normalized Lebesgue measure on the $U$-orbit $a(\\hat s) u([1])x_0$ removed an exponentially small proportion. Suppose $\\hat \\mu$ is $(4-\\theta, s')$-good. Then $a(2s')_\\ast \\hat \\mu$ can be written as a(2s')_\\ast \\hat \\mu = \\nu_{\\S} \\mu_{\\S} + \\sum_{k} \\nu_k a(2s')_\\ast \\mu_k, where $\\mu_{\\S}$ is a probability measure on $X$, $\\nu_{\\S} = O(\\beta)$, and for each $k$, $\\mu_k$ is a probability measure on $$\\exp\\left(B_{\\vv r^+}(\\beta)\\right)u\\left([\\beta]\\right)x_k$$ for some $x_k \\in X$. Moreover, for any $f \\in C_c^{\\infty}(X)$, \\[ \\int f \\dd \\mu_k = \\int_{[\\beta]^2} \\int_{[1]} f(\\exp(v_1 \\vv v_1 + w_1\\vv w_1 )u(\\beta r)) \\dd r \\dd \\rho_k (v_1, w_1),\\] where $\\rho_k$ is a probability measure on $[\\beta]^2$ with dimension larger than $2-2\\theta$ at scale $s'$ (see Proposition \\ref{prop:high-dimension-to-equidistribution-a} for the definition).", "argument_gist": "Conditions a high-dimensional orbit measure on anisotropic boxes, projects away contracting and block-diagonal coordinates, and seeks a transverse Frostman measure to which Proposition 3.1 applies.", "gist_origin": "review_model", "proof_anchor": { "source": "manuscript", "line_start": 3260, "line_end": 3340, "char_start": 197404, "char_end": 202755 }, "proof_opening": "\\par Let us define \\Upsilon_{s'} := B^{A_0 A U^\\ast}(\\beta) \\exp\\left(B_{\\vv r^-} (\\beta e^{-s'})\\right)\\exp\\left(B_{\\vv r^+} (\\beta)\\right) B^{U}(\\beta), and decompose $X_{\\beta^{1/4}}$ as \\[ X_{\\beta^{1/4}} = \\bigcup_{k=1}^{\\beta^{-100} e^{2s'}} \\Upsilon_{s'} x_k. \\] We say $x_k$ is light if \\[ \\hat \\mu\\left( \\Upsilon_{s'} x_k\\right) \\le \\beta^{-2} e^{-2s' -\\theta s'}. \\] Otherwise we say $x_k$ is heavy. For each heavy $x_k$, let us denote \\[ \\bar{\\nu}_k = \\hat \\mu \\left( \\Upsilon_{s'} x_k \\right), \\] and \\[ \\bar{\\mu}_k = \\bar{\\nu}^{-1}_k \\hat \\mu|_{ \\Upsilon_{s'} x_k}.\\] Let us define \\[ \\Upsilon_{\\mathrm{light}} := \\bigcup_{x_k \\text{ light }} \\Upsilon_{s'} x_k.\\] Then it is easy to see ", "anchor": { "source": "manuscript", "line_start": 3243, "line_end": 3260, "char_start": 196265, "char_end": 197403 }, "review": { "verdict": "major_issues", "issues": [ "Equation eq:decompose-hat-mu is an exact positive-measure decomposition, but the proof replaces each component by a measure only O(beta)-close against C^1 tests. A weak signed error cannot be represented without proof as a small positive probability component.", "The Upsilon sets form a cover, not a partition. Summing their normalized restrictions with weights equal to their masses overcounts overlaps in eq:first-decompose-hat-mu.", "An arbitrary exponentially small deletion from an orbit need not leave a union of complete beta-length U-segments. The orbitwise construction and resulting transverse weights are not defined rigorously.", "The displayed formulas for integrating f against mu_k omit the point x_k, so f is incorrectly evaluated on a group element rather than a point of X.", "The pointwise comparison before eq:compare-u-orbit omits a(2s') on its right-hand orbit.", "Transporting the measure does not preserve the exact niceness bound without controlling preimages of enlarged neighborhoods.", "The base points must satisfy the thick-part hypothesis of Proposition 3.1; this is not stated in the proposition's conclusion." ], "reasoning": "The proof outlines a plausible weak approximation strategy, not the exact convex decomposition stated. A corrected theorem would need a precise test-function error, a genuine disintegration or partition, and quantitative control of the transported Frostman bounds." } }, { "id": "manuscript", "kind": "paper", "label": "Effective version of Ratner's equidistribution theorem for SL(3,R)", "argument_gist": "The target is uniform effective equidistribution of a long E_23-unipotent segment unless a prescribed central diagonal translate has an exceptionally short vector or bivector. The proposed proof first obtains weak transverse dimension, then repeatedly increases it through a weighted Kakeya model. Random and intermediate-scale concentrated packets are treated by intersection counts; the structured case is meant to force concentration near an SL(2,R) semidirect R^2 orbit, excluded by the Diophantine alternative. A final projection to the expanding transverse plane and a van der Corput estimate should give equidistribution. The bottlenecks are the initial closing argument, the encounter-hyperbola geometry and weighted counting, and the final measure projection; none is fully established in the supplied text.", "anchor": { "source": "manuscript", "line_start": 1, "line_end": 3562, "char_start": 0, "char_end": 211793 }, "gist_origin": "review_model" }, { "id": "subsec-notation", "kind": "anchor", "label": "subsec-notation", "owner": "sec-intro", "section": "sec-intro", "anchor": { "source": "manuscript", "line_start": 434, "line_end": 435, "char_start": 15142, "char_end": 15165 }, "argument_gist": "", "excerpt": "bits in $G/\\Gamma$ for $G = \\SL(2, \\R) \\times \\SL(2, \\R)$ or $\\SL(2, \\C)$. \\subsection{Notation} \\par Throughout this paper we will fix the following notation. \\item For a normed vector space $V$ and $r >0$, let $B_V(r)$ denote the set of vectors in $V$ with norm $\\le r$. \\item For a Lie group $J$ and $r >0$, let $B^J(r)$ denote the $r$-neighborhood of the identity in $J$. \\" }, { "id": "subsec-main-results", "kind": "anchor", "label": "subsec-main-results", "owner": "sec-intro", "section": "sec-intro", "anchor": { "source": "manuscript", "line_start": 449, "line_end": 450, "char_start": 16097, "char_end": 16124 }, "argument_gist": "", "excerpt": "or $L >0$, let $[L]$ denote the interval $[-L/2, L/2]$. \\subsection{Main results} \\par In this paper we will focus on the following case: Let $G = \\SL(3,\\R)$, $\\Gamma = \\SL(3,\\Z)$, $X = G/\\Gamma$. Let $U = \\{u(r): r\\in \\R \\}$ be a one-parameter unipotent subgroup of $G$ defined as follows: u(r) := 1 & 0 & 0 \\\\ 0 & 1 & r" }, { "id": "eq:def-U", "kind": "anchor", "label": "eq:def-U", "owner": "sec-intro", "section": "sec-intro", "anchor": { "source": "manuscript", "line_start": 453, "line_end": 454, "char_start": 16354, "char_end": 16370 }, "argument_gist": "", "excerpt": "r): r\\in \\R \\}$ be a one-parameter unipotent subgroup of $G$ defined as follows: u(r) := 1 & 0 & 0 \\\\ 0 & 1 & r \\\\ 0 & 0 & 1 . Let $\\mu_G$ denote the unique $G$-invariant probability measure on $X$. \\par For $\\varpi > 0$, let us define X_\\varpi := \\left\\{ x = g\\Gamma \\in X: \\|g" }, { "id": "eq:def-compact-set", "kind": "anchor", "label": "eq:def-compact-set", "owner": "sec-intro", "section": "sec-intro", "anchor": { "source": "manuscript", "line_start": 461, "line_end": 462, "char_start": 16599, "char_end": 16625 }, "argument_gist": "", "excerpt": "$G$-invariant probability measure on $X$. \\par For $\\varpi > 0$, let us define X_\\varpi := \\left\\{ x = g\\Gamma \\in X: \\|g \\vv v\\| \\ge \\varpi \\text{, } \\forall \\vv v \\in \\Z^3\\setminus\\{\\vv 0\\} \\text{ or } \\wedge^2 \\Z^3\\setminus\\{\\vv 0\\}. \\right\\} \\par By Mahler's criterion, every compact subset of $X$ is contained in some $X_\\varpi$. \\par For $t \\in \\R$, let" }, { "id": "eq:define-a", "kind": "anchor", "label": "eq:define-a", "owner": "sec-intro", "section": "sec-intro", "anchor": { "source": "manuscript", "line_start": 467, "line_end": 468, "char_start": 16955, "char_end": 16974 }, "argument_gist": "", "excerpt": "bset of $X$ is contained in some $X_\\varpi$. \\par For $t \\in \\R$, let us denote a(t) := 1 & & \\\\ & e^{t/2} & \\\\ & & e^{-t/2} \\in G, and a_0(t) := e^{t/3} & & \\\\ & e^{-t/6} & \\\\ & & e^{-t/6} \\in G. We will prove the following result" }, { "id": "eq:define-a0", "kind": "anchor", "label": "eq:define-a0", "owner": "sec-intro", "section": "sec-intro", "anchor": { "source": "manuscript", "line_start": 474, "line_end": 475, "char_start": 17104, "char_end": 17124 }, "argument_gist": "", "excerpt": "1 & & \\\\ & e^{t/2} & \\\\ & & e^{-t/2} \\in G, and a_0(t) := e^{t/3} & & \\\\ & e^{-t/6} & \\\\ & & e^{-t/6} \\in G. We will prove the following result concerning effective equidistribution of $U$-orbit: There exist $\\eta >0$ and $t_0 >1$ such that for any" }, { "id": "eq:def-H", "kind": "anchor", "label": "eq:def-H", "owner": "sec-prelim", "section": "sec-prelim", "anchor": { "source": "manuscript", "line_start": 630, "line_end": 631, "char_start": 25494, "char_end": 25510 }, "argument_gist": "", "excerpt": "em. \\par Let $G = \\SL(3,\\R)$, and $H \\subset G$ be the following subgroup of $G$: H := \\left\\{ 1 & \\\\ & h \\in G : h \\in \\SL(2,\\R) \\right\\}. Clearly $H$ is isomorphic to $\\SL(2,\\R)$. \\par Let $\\fg$ and $\\fh$ denote the Lie algebras of $G$ and $H$, respectively. \\par Consider the adjoit action of $H$ on $\\fg$, we have the following dec" }, { "id": "eq:decomp-g", "kind": "anchor", "label": "eq:decomp-g", "owner": "sec-prelim", "section": "sec-prelim", "anchor": { "source": "manuscript", "line_start": 637, "line_end": 638, "char_start": 25841, "char_end": 25860 }, "argument_gist": "", "excerpt": "Consider the adjoit action of $H$ on $\\fg$, we have the following decomposition: \\fg = \\fh + \\fr_0 + \\fr_1 + \\fr_2, where $\\fr_0, \\fr_1, \\fr_2$ are irreducible subrepresentations with respect to the adjoint action of $H$, and $\\dim \\fr_1 = \\dim \\fr_2 =2$, $\\dim \\fr_0 =1$. In particular, $\\fr_0 = \\R \\fa_0$ where \\[ \\fa_0 = 1/3 & 0 & 0 \\\\ 0 &" }, { "id": "eq:def-b", "kind": "anchor", "label": "eq:def-b", "owner": "sec-prelim", "section": "sec-prelim", "anchor": { "source": "manuscript", "line_start": 707, "line_end": 708, "char_start": 28124, "char_end": 28140 }, "argument_gist": "", "excerpt": "\\[ \\Ad (a_0(t)) \\bw = e^{t/2} \\bw, \\text{ for } \\bw \\in \\fr_2. \\] Let us denote b(t) := a(t)a_0(-t) = e^{-t/3} & & \\\\ & e^{2t/3} & \\\\ & & e^{-t/3} \\in G, and a_1(t) := a(t)a_0(t) = e^{t/3} & & \\\\ & e^{t/3} & \\\\ & & e^{-2t/3} \\in G. \\par Let u" }, { "id": "eq:def-a1", "kind": "anchor", "label": "eq:def-a1", "owner": "sec-prelim", "section": "sec-prelim", "anchor": { "source": "manuscript", "line_start": 711, "line_end": 712, "char_start": 28285, "char_end": 28302 }, "argument_gist": "", "excerpt": "t/3} & & \\\\ & e^{2t/3} & \\\\ & & e^{-t/3} \\in G, and a_1(t) := a(t)a_0(t) = e^{t/3} & & \\\\ & e^{t/3} & \\\\ & & e^{-2t/3} \\in G. \\par Let us denote \\vv r^+ := \\R \\vv w_1 + \\R \\vv v_1, and \\vv r^-:= \\R \\vv w_2 + \\R" }, { "id": "eq:r+", "kind": "anchor", "label": "eq:r+", "owner": "sec-prelim", "section": "sec-prelim", "anchor": { "source": "manuscript", "line_start": 718, "line_end": 719, "char_start": 28469, "char_end": 28482 }, "argument_gist": "", "excerpt": "/3} & \\\\ & & e^{-2t/3} \\in G. \\par Let us denote \\vv r^+ := \\R \\vv w_1 + \\R \\vv v_1, and \\vv r^-:= \\R \\vv w_2 + \\R \\vv v_2, Let us denote by $p_{1,2}: \\fg \\to \\fr_1 + \\fr_2$ the projection of $\\fg$ to $\\fr_1 + \\fr_2$. For $j=0, 1,2$, let $p_j : \\fg \\to \\fr_j$ denote the projecti" }, { "id": "eq:r-", "kind": "anchor", "label": "eq:r-", "owner": "sec-prelim", "section": "sec-prelim", "anchor": { "source": "manuscript", "line_start": 723, "line_end": 724, "char_start": 28560, "char_end": 28573 }, "argument_gist": "", "excerpt": "ation} \\vv r^+ := \\R \\vv w_1 + \\R \\vv v_1, and \\vv r^-:= \\R \\vv w_2 + \\R \\vv v_2, Let us denote by $p_{1,2}: \\fg \\to \\fr_1 + \\fr_2$ the projection of $\\fg$ to $\\fr_1 + \\fr_2$. For $j=0, 1,2$, let $p_j : \\fg \\to \\fr_j$ denote the projection from $\\fg$ to $\\fr_j$. Let $p_+$, $p_-$, $p_{\\vv w_j}$, $p_{\\vv v_j}$ and $p_{\\fu^\\ast" }, { "id": "eq:product-w-h-1", "kind": "anchor", "label": "eq:product-w-h-1", "owner": "lm:product-lie-algebra-1", "section": "sec-prelim", "anchor": { "source": "manuscript", "line_start": 759, "line_end": 760, "char_start": 30315, "char_end": 30339 }, "argument_gist": "", "excerpt": "r_2}(\\rho_1)$ and $\\vv h \\in B_{\\R \\fa_0 + \\fh}(\\rho_2)$, we have that \\exp(\\vv w) \\exp(\\vv h) = \\exp(\\tilde{\\vv w} +\\tilde{\\vv h}), where $\\tilde{\\vv w} \\in \\fr_1 + \\fr_2$ with $\\|\\vv w - \\tilde{\\vv w}\\| \\le 2\\rho_2 \\|\\vv w\\|$, and $\\tilde{\\vv h} \\in \\R \\fa_0 + \\fh$ with $$\\|\\vv h - \\tilde{\\vv h} \\| \\le 2\\rho_1 \\|\\vv h\\|.$$ \\" }, { "id": "eq:product-w-h-2", "kind": "anchor", "label": "eq:product-w-h-2", "owner": "lm:product-lie-algebra-1", "section": "sec-prelim", "anchor": { "source": "manuscript", "line_start": 766, "line_end": 767, "char_start": 30794, "char_end": 30818 }, "argument_gist": "", "excerpt": "}(\\rho_1)$ and $\\tilde{\\vv h} \\in B_{\\R \\fa_0 + \\fh}(\\rho_2)$, we have \\exp(\\tilde{\\vv w} +\\tilde{\\vv h}) = \\exp(\\vv w) \\exp(\\vv h), where $\\vv w \\in \\fr_1 + \\fr_2$ with $\\|\\vv w - \\tilde{\\vv w}\\| \\le 4\\rho_2 \\|\\tilde {\\vv w}\\|$, and $\\vv h \\in \\R \\fa_0 + \\fh$ with $$\\|\\vv h - \\tilde{\\vv h} \\| \\le 4\\rho_1 \\|\\tilde{\\vv h}\\|.$$ \\end" }, { "id": "eq:hatw-ww'", "kind": "anchor", "label": "eq:hatw-ww'", "owner": "lm:product-w", "section": "sec-prelim", "anchor": { "source": "manuscript", "line_start": 828, "line_end": 829, "char_start": 35387, "char_end": 35406 }, "argument_gist": "", "excerpt": "\\] where $\\hat{\\vv w} = \\vv w + \\vv w' + \\bar{\\vv w}$. In particular, \\| \\hat{\\vv w} - \\vv w - \\vv w'\\| \\le 2 \\rho \\rho_0. It is easy to see that \\[ \\|\\hat{\\vv w}\\| \\le 3 \\rho_0. \\] By Lemma \\ref{lm:product-lie-algebra-1}, we have that \\[ \\exp(\\hat{\\vv w} + \\bar{\\vv h}) = \\exp(\\tilde{\\vv w}) \\exp(\\vv h) \\] where $\\tilde{\\v" }, { "id": "eq:tildew-hatw", "kind": "anchor", "label": "eq:tildew-hatw", "owner": "lm:product-w", "section": "sec-prelim", "anchor": { "source": "manuscript", "line_start": 837, "line_end": 838, "char_start": 35766, "char_end": 35788 }, "argument_gist": "", "excerpt": "w}) \\exp(\\vv h) \\] where $\\tilde{\\vv w} \\in \\fr_1 + \\fr_2$ satisfying \\| \\tilde{\\vv w} - \\hat{\\vv w} \\| \\le 8 \\rho \\|\\hat{\\vv w}\\| \\le 24 \\rho_0 \\rho, and $\\vv h \\in \\R \\fa_0 + \\fh$ with $\\|\\vv h\\| \\le 4 \\rho$. Noting that \\eqref{eq:hatw-ww'} and \\eqref{eq:tildew-hatw} imply that \\[ \\| \\tilde{\\vv w} - \\vv w - \\vv w'\\| \\le 26 \\rho_0 \\rh" }, { "id": "subsec-outline", "kind": "anchor", "label": "subsec-outline", "owner": "sec-prelim", "section": "sec-prelim", "anchor": { "source": "manuscript", "line_start": 847, "line_end": 848, "char_start": 36174, "char_end": 36196 }, "argument_gist": "", "excerpt": "26 \\rho_0 \\rho, \\] we conclude the lemma. \\subsection{Outline of the proof} \\par In this subsection we will give the outline of the proof. \\par The proof is inspired by Ratner's original proof of her measure rigidity theorem \\cite{ratner-acta} and recent papers by Lindenstrauss, Mohammadi, and Wang \\cite{LM-preprint2021,LMW-preprint2022}. However, compared with prev" }, { "id": "subsec-initial-construction", "kind": "anchor", "label": "subsec-initial-construction", "owner": "sec-initial-dim-control", "section": "sec-initial-dim-control", "anchor": { "source": "manuscript", "line_start": 998, "line_end": 999, "char_start": 47277, "char_end": 47312 }, "argument_gist": "", "excerpt": "on Control} \\subsection{Initial construction of transversal sets} \\par We first introduce some notation. \\par For $\\ell >1$ and a subspace $V \\subset \\fr_1 + \\fr_2$ generated by some vectors from the canonical basis of $\\fr_1 + \\fr_2$, namely, $\\{ \\vv v_{1}, \\vv v_2, \\vv w_1, \\vv w_{2}\\}$, let us define Q(V, \\ell, \\bar \\" }, { "id": "eq:def-thin", "kind": "anchor", "label": "eq:def-thin", "owner": "sec-initial-dim-control", "section": "sec-initial-dim-control", "anchor": { "source": "manuscript", "line_start": 1004, "line_end": 1005, "char_start": 47573, "char_end": 47592 }, "argument_gist": "", "excerpt": "+ \\fr_2$, namely, $\\{ \\vv v_{1}, \\vv v_2, \\vv w_1, \\vv w_{2}\\}$, let us define Q(V, \\ell, \\bar \\beta) := \\{\\exp(\\vv w): \\vv w \\in B_{\\fr_1 + \\fr_2}(\\bar \\beta), \\|p_{V}(\\vv w)\\|\\le \\bar \\beta e^{-\\ell}\\}, where $p_V$ denotes the projection to $V$ with respect to the canonical basis. \\par Let us denote { \\fq^{A_0 H}(\\ell, \\bar" }, { "id": "eq:def-qH", "kind": "anchor", "label": "eq:def-qH", "owner": "sec-initial-dim-control", "section": "sec-initial-dim-control", "anchor": { "source": "manuscript", "line_start": 1011, "line_end": 1012, "char_start": 47852, "char_end": 47869 }, "argument_gist": "", "excerpt": "tes the projection to $V$ with respect to the canonical basis. \\par Let us denote { \\fq^{A_0 H}(\\ell, \\bar \\beta )&:= \\left\\{ \\vv h \\in B_{\\R \\fa_0 + \\fh} (\\bar \\beta): \\| p_{\\fu^\\ast}(\\vv h) \\| \\le \\bar \\beta e^{-\\ell} \\right\\}, \\\\ Q^{A_0 H}(\\ell, \\bar \\beta) &:= \\left\\{ h = \\exp(\\vv h): \\vv h \\in \\fq^{A_0 H}(\\ell, \\bar \\beta) \\right\\}, and \\begin{" }, { "id": "eq:def-QH", "kind": "anchor", "label": "eq:def-QH", "owner": "sec-initial-dim-control", "section": "sec-initial-dim-control", "anchor": { "source": "manuscript", "line_start": 1013, "line_end": 1014, "char_start": 48020, "char_end": 48037 }, "argument_gist": "", "excerpt": "n B_{\\R \\fa_0 + \\fh} (\\bar \\beta): \\| p_{\\fu^\\ast}(\\vv h) \\| \\le \\bar \\beta e^{-\\ell} \\right\\}, \\\\ Q^{A_0 H}(\\ell, \\bar \\beta) &:= \\left\\{ h = \\exp(\\vv h): \\vv h \\in \\fq^{A_0 H}(\\ell, \\bar \\beta) \\right\\}, and \\fr^{A_0 H}(\\ell, \\bar \\beta) &:= \\left\\{ \\vv h \\in B_{\\R \\fa_0 + \\fh} (\\bar \\beta): \\| p_{\\fu}(\\vv h) \\| \\le \\bar \\beta e^{-\\ell} \\right\\}, \\\\" }, { "id": "eq:def-rH", "kind": "anchor", "label": "eq:def-rH", "owner": "sec-initial-dim-control", "section": "sec-initial-dim-control", "anchor": { "source": "manuscript", "line_start": 1018, "line_end": 1019, "char_start": 48176, "char_end": 48193 }, "argument_gist": "", "excerpt": "{ h = \\exp(\\vv h): \\vv h \\in \\fq^{A_0 H}(\\ell, \\bar \\beta) \\right\\}, and \\fr^{A_0 H}(\\ell, \\bar \\beta) &:= \\left\\{ \\vv h \\in B_{\\R \\fa_0 + \\fh} (\\bar \\beta): \\| p_{\\fu}(\\vv h) \\| \\le \\bar \\beta e^{-\\ell} \\right\\}, \\\\ R^{A_0 H}(\\ell, \\bar \\beta) &:= \\left\\{ h = \\exp(\\vv h) : \\vv h \\in \\fr^{A_0 H}(\\ell, \\bar \\beta) \\right\\}. } \\par For $" }, { "id": "eq:def-RH", "kind": "anchor", "label": "eq:def-RH", "owner": "sec-initial-dim-control", "section": "sec-initial-dim-control", "anchor": { "source": "manuscript", "line_start": 1020, "line_end": 1021, "char_start": 48340, "char_end": 48357 }, "argument_gist": "", "excerpt": "h \\in B_{\\R \\fa_0 + \\fh} (\\bar \\beta): \\| p_{\\fu}(\\vv h) \\| \\le \\bar \\beta e^{-\\ell} \\right\\}, \\\\ R^{A_0 H}(\\ell, \\bar \\beta) &:= \\left\\{ h = \\exp(\\vv h) : \\vv h \\in \\fr^{A_0 H}(\\ell, \\bar \\beta) \\right\\}. } \\par For $s_1, s_2 \\ge 0$, $\\bar \\beta >0$, let us denote Q^{s_1}_{ s_2}(\\bar \\beta) := \\{\\exp(\\vv w): \\vv w \\in \\fr_1 + \\fr_2 : \\|p_+" }, { "id": "eq:def-Q", "kind": "anchor", "label": "eq:def-Q", "owner": "sec-initial-dim-control", "section": "sec-initial-dim-control", "anchor": { "source": "manuscript", "line_start": 1026, "line_end": 1026, "char_start": 48561, "char_end": 48577 }, "argument_gist": "", "excerpt": "ight\\}. } \\par For $s_1, s_2 \\ge 0$, $\\bar \\beta >0$, let us denote Q^{s_1}_{ s_2}(\\bar \\beta) := \\{\\exp(\\vv w): \\vv w \\in \\fr_1 + \\fr_2 : \\|p_+(\\vv w)\\| \\le \\bar \\beta e^{-s_1}, \\|p_- (\\vv w)\\| \\le \\bar \\beta e^{-s_2} \\}, and $Q_s(\\bar \\beta) := Q_s^s(\\bar \\beta)$. \\par Let $x_{0} \\in X$ be such that (2) in Theorem \\ref{thm:main-thm} does not ho" }, { "id": "eq:decomposition-measure", "kind": "anchor", "label": "eq:decomposition-measure", "owner": "sec-initial-dim-control", "section": "sec-initial-dim-control", "anchor": { "source": "manuscript", "line_start": 1043, "line_end": 1044, "char_start": 49653, "char_end": 49685 }, "argument_gist": "", "excerpt": "to show that the removed proportion is $O(s^2 \\beta^{ \\alpha_2})$. Therefore, \\mu_{\\cE} = \\mu_{\\cF} + O(s^2 \\beta^{ \\alpha_2}), where $\\mu_{\\cE}$ denotes the normalized $U$-invariant measure on $\\cE$. \\subsection{Initial dimension bound} \\par We shall prove the following proposition: \\label{prop:initial-bound-" }, { "id": "subsec-initialize", "kind": "anchor", "label": "subsec-initialize", "owner": "sec-initial-dim-control", "section": "sec-initial-dim-control", "anchor": { "source": "manuscript", "line_start": 1050, "line_end": 1051, "char_start": 49870, "char_end": 49895 }, "argument_gist": "", "excerpt": "\\cE}$ denotes the normalized $U$-invariant measure on $\\cE$. \\subsection{Initial dimension bound} \\par We shall prove the following proposition: For any $\\ell \\in [\\delta_3 t, \\delta_1 t]$, and any $x \\in \\cF$, we have \\[ \\mu_{\\cF} (Q_\\ell(\\beta) B^{A_0 H}(\\beta) x) \\le e^{- d_0 \\ell}. \\] \\par Let $\\eta_1" }, { "id": "eq:a0ge", "kind": "anchor", "label": "eq:a0ge", "owner": "prop:initial-bound-dimension", "section": "sec-initial-dim-control", "anchor": { "source": "manuscript", "line_start": 1074, "line_end": 1075, "char_start": 51355, "char_end": 51370 }, "argument_gist": "", "excerpt": "d \\[ \\vv h_{\\bar x} = a_0 \\fa + a_1 \\fu + a_2 \\fu^\\ast,\\] we will have |a_0| \\ge \\beta e^{-2 d_0 \\ell} \\par Let us write $\\bar x = u_{\\bar x} x$ where $u_{\\bar x} = u(r e^{\\eta_1 \\ell})$ for some $r \\in [1]$. Let us fix a representative $g \\in G$ of $x$, then we have \\[ u(r_{\\bar x} e^{ d_0 \\ell}) u_{\\bar x} g = \\exp (\\vv v_{\\bar x} ) a^{\\bar" }, { "id": "eq:bound-2", "kind": "anchor", "label": "eq:bound-2", "owner": "prop:initial-bound-dimension", "section": "sec-initial-dim-control", "anchor": { "source": "manuscript", "line_start": 1082, "line_end": 1083, "char_start": 51823, "char_end": 51841 }, "argument_gist": "", "excerpt": "r some $\\gamma_{\\bar x} \\in \\Gamma$. The above equality is equivalent to g \\gamma_{\\bar x} g^{-1} = (a_0^{\\bar x} h_{\\bar x} u_{\\bar x})^{-1} \\exp(-\\vv v_{\\bar x}) u(r_{\\bar x} e^{2 d_0 \\ell}) u_{\\bar x}. \\par Without loss of generality, we can assume that for different $\\bar x_1$ and $\\bar x_2$ as above, $|u^{-1}_{\\bar x_1} u_{\\bar x_2}|" }, { "id": "eq:bound-3", "kind": "anchor", "label": "eq:bound-3", "owner": "prop:initial-bound-dimension", "section": "sec-initial-dim-control", "anchor": { "source": "manuscript", "line_start": 1104, "line_end": 1105, "char_start": 53535, "char_end": 53553 }, "argument_gist": "", "excerpt": "-1} u_{\\bar x'} u(r_{\\bar x'} e^{2 d_0 \\ell}).\\] This implies that u(-L) h_{\\bar x} u(L) = u(r e^{2 d_0 \\ell}) h_{\\bar x'}, where $u(L) = u_{\\bar x} u^{-1}_{\\bar x'}$ and $r = r_{\\bar x} - r'_{\\bar x'}$. Note that $|r|\\le 1$ and $|L| \\ge e^{10 d_0 \\ell}$. $|r| \\le 1$ implies that the norm of the right hand side is $\\le e^{2 d_0 \\ell}$." }, { "id": "eq:pi-s-l", "kind": "anchor", "label": "eq:pi-s-l", "owner": "sec-reduction", "section": "sec-reduction", "anchor": { "source": "manuscript", "line_start": 1222, "line_end": 1223, "char_start": 62767, "char_end": 62784 }, "argument_gist": "", "excerpt": "cF_i}$. \\par For notational simplicity, for $s' >0, \\ell \\ge 0$, let us denote \\Pi_{s', \\ell} := a(2s')B^{A_0 A}(\\beta) B^{U^\\ast}(\\beta)\\exp\\left( B_{\\vv r^-}(\\beta e^{-\\ell}) \\right) \\exp\\left(B_{\\vv r^+}(\\beta e^{-\\ell})\\right) B^{U}(\\beta)a(-2s'), and \\Pi_{s'} := \\Pi_{s', 0}. \\par For $x \\in X$, let us" }, { "id": "eq:pi-s", "kind": "anchor", "label": "eq:pi-s", "owner": "sec-reduction", "section": "sec-reduction", "anchor": { "source": "manuscript", "line_start": 1227, "line_end": 1228, "char_start": 62999, "char_end": 63014 }, "argument_gist": "", "excerpt": "eft(B_{\\vv r^+}(\\beta e^{-\\ell})\\right) B^{U}(\\beta)a(-2s'), and \\Pi_{s'} := \\Pi_{s', 0}. \\par For $x \\in X$, let us denote \\[ \\Pi(x) := \\Pi_{s'} x. \\] \\par Given that $\\mu_{\\cF_i}$ is $(d_i, s_i)$-good, our aim is to prove that $\\mu = a(s_i)_\\ast \\mu_{\\cF_i}$ is $(d_i + \\epsilon_2, s_{i+1})$-good after removing an exponentially small propor" }, { "id": "eq:omega-neighborhood", "kind": "anchor", "label": "eq:omega-neighborhood", "owner": "sec-kakeya-model", "section": "sec-kakeya-model", "anchor": { "source": "manuscript", "line_start": 1293, "line_end": 1293, "char_start": 67064, "char_end": 67093 }, "argument_gist": "", "excerpt": "n of the orbit measure. \\par Let us introduce some notation. \\par Let us denote \\Omega_{s'} := Q^{s'}_{3s'}(2\\beta) Q^{A_0 H}(2s', 2\\beta) , and \\Theta_{s'} := B^{A_0 A}(\\beta) B^{U^\\ast}(\\beta e^{-2s'})\\exp\\left( B_{\\vv r^-}(\\beta e^{-s'}) \\right)\\exp\\left( B_{\\vv r^+}(\\beta e^{-s'}) \\right) B^U(\\beta)." }, { "id": "eq:theta-neighborhood", "kind": "anchor", "label": "eq:theta-neighborhood", "owner": "sec-kakeya-model", "section": "sec-kakeya-model", "anchor": { "source": "manuscript", "line_start": 1295, "line_end": 1296, "char_start": 67188, "char_end": 67217 }, "argument_gist": "", "excerpt": "hood}\\Omega_{s'} := Q^{s'}_{3s'}(2\\beta) Q^{A_0 H}(2s', 2\\beta) , and \\Theta_{s'} := B^{A_0 A}(\\beta) B^{U^\\ast}(\\beta e^{-2s'})\\exp\\left( B_{\\vv r^-}(\\beta e^{-s'}) \\right)\\exp\\left( B_{\\vv r^+}(\\beta e^{-s'}) \\right) B^U(\\beta)." }, { "id": "eq:sigma-neighborhood", "kind": "anchor", "label": "eq:sigma-neighborhood", "owner": "sec-kakeya-model", "section": "sec-kakeya-model", "anchor": { "source": "manuscript", "line_start": 1304, "line_end": 1305, "char_start": 67580, "char_end": 67609 }, "argument_gist": "", "excerpt": "For $s' >0$, let us denote \\Sigma_{s'} := B^{A_0 A}(\\beta) B^{U^\\ast}(\\beta e^{-2s'}) B_{\\vv r^-}(\\beta e^{-s'})B_{\\vv r^+}(\\beta e^{s'}) B^U(\\beta). For $x \\in X$, let us denote \\[ \\Sigma(x) := \\Sigma_{s'} x,\\] \\[ \\Theta(x) := \\Theta_{s'} x, \\] and \\[ \\O" }, { "id": "eq:cI", "kind": "anchor", "label": "eq:cI", "owner": "def:kakeya-model", "section": "sec-kakeya-model", "anchor": { "source": "manuscript", "line_start": 1471, "line_end": 1472, "char_start": 77779, "char_end": 77792 }, "argument_gist": "", "excerpt": "right]^2. \\] \\par Let $\\cI$ be a finite subset of $\\Pi(x)$ defined as follows: \\cI := \\left\\{ \\exp(v_1 \\fd \\vv v_1 + w_1 \\fd \\vv w_1 ) u(r \\fd') x: r, v_1, w_1 \\in \\left[10 e^{2s'}\\right]\\cap \\Z \\right\\}, where $\\fd = 0.1 \\beta e^{-s'}, \\fd' = 0.1 \\beta$. For $z \\in \\cI$, let $\\Theta'(z) := \\Theta(z) \\cap \\Pi(x)$." }, { "id": "eq:cIz", "kind": "anchor", "label": "eq:cIz", "owner": "def:kakeya-model", "section": "sec-kakeya-model", "anchor": { "source": "manuscript", "line_start": 1480, "line_end": 1481, "char_start": 78474, "char_end": 78488 }, "argument_gist": "", "excerpt": "$z \\in \\cI$, let $\\cI_z$ be a finite subset of $\\Theta'(z)$ defined as follows: \\cI_z := \\left\\{ \\exp(v_2 \\fd'' \\vv v_2 + w_2 \\fd'' \\vv w_2 ) z : v_2, w_2 \\in \\left[10 e^{2s'}\\right]\\cap \\Z \\right\\}, where $\\fd'' = 0.1 \\beta e^{-3s'}$. For $y \\in \\cI_z$, let $\\Omega'(y):= \\Omega(y) \\cap \\Theta'(z)$. Let us denote $$\\fJ := \\bigcup_{z \\in \\cI} \\cI_z.$$" }, { "id": "eq:fly", "kind": "anchor", "label": "eq:fly", "owner": "def:kakeya-model", "section": "sec-kakeya-model", "anchor": { "source": "manuscript", "line_start": 1495, "line_end": 1496, "char_start": 79536, "char_end": 79550 }, "argument_gist": "", "excerpt": "\\par For any $y \\in \\Pi(x)$, let us define \\fl(y) := \\left\\{ r \\in \\left[2\\beta e^{2s'}\\right]: u(r) y \\in \\Pi(x) \\right\\}. \\par For each $z \\in \\cI$ and $y \\in \\cI_z$, let us write \\[ y = \\exp\\left(\\vv w^-_y\\right) \\exp\\left(\\vv w^+_y\\right) u\\left(r_y \\right) x,\\] where $$\\vv w^-_y = v_{2, y} \\vv v_2 + w_{2, y} \\vv" }, { "id": "eq:fvy", "kind": "anchor", "label": "eq:fvy", "owner": "def:kakeya-model", "section": "sec-kakeya-model", "anchor": { "source": "manuscript", "line_start": 1507, "line_end": 1508, "char_start": 80267, "char_end": 80281 }, "argument_gist": "", "excerpt": "u(r) y\\right) : r \\in \\fl(y) \\right\\}. \\] It is straightforward to check that f_{v, y}(r) = \\frac{2r v_{2, y}}{2 + r v_{2, y} w_{2, y}} + v_{1, y}, f_{w, y}(r) = \\frac{-2r w_{2, y}}{2 - r v_{2, y} w_{2, y}} + w_{1, y}, and {rl} f_y(r) & = r_y + \\frac{" }, { "id": "eq:fwy", "kind": "anchor", "label": "eq:fwy", "owner": "def:kakeya-model", "section": "sec-kakeya-model", "anchor": { "source": "manuscript", "line_start": 1511, "line_end": 1512, "char_start": 80391, "char_end": 80405 }, "argument_gist": "", "excerpt": "}(r) = \\frac{2r v_{2, y}}{2 + r v_{2, y} w_{2, y}} + v_{1, y}, f_{w, y}(r) = \\frac{-2r w_{2, y}}{2 - r v_{2, y} w_{2, y}} + w_{1, y}, and {rl} f_y(r) & = r_y + \\frac{2r + 2r w_{1,y} v_{2, y}}{2 + r v_{2, y} w_{2, y}} + \\frac{w_{1,y}v_{1,y} - f_{v, y}(r)f_{w, y}(r)}{2} \\\\ &= r_y" }, { "id": "eq:fy", "kind": "anchor", "label": "eq:fy", "owner": "def:kakeya-model", "section": "sec-kakeya-model", "anchor": { "source": "manuscript", "line_start": 1516, "line_end": 1517, "char_start": 80521, "char_end": 80534 }, "argument_gist": "", "excerpt": "\\frac{-2r w_{2, y}}{2 - r v_{2, y} w_{2, y}} + w_{1, y}, and {rl} f_y(r) & = r_y + \\frac{2r + 2r w_{1,y} v_{2, y}}{2 + r v_{2, y} w_{2, y}} + \\frac{w_{1,y}v_{1,y} - f_{v, y}(r)f_{w, y}(r)}{2} \\\\ &= r_y + r + \\frac{r w_{2,y} v_{1,y}}{2 - r w_{2,y} v_{2,y}} + \\frac{r w_{1,y} v_{2,y}}{2 + r w_{2,y} v_{2,y}} + \\frac{ r (r w_{2,y} v_" }, { "id": "eq:delta-y_1-y_2", "kind": "anchor", "label": "eq:delta-y_1-y_2", "owner": "def:kakeya-model", "section": "sec-kakeya-model", "anchor": { "source": "manuscript", "line_start": 1533, "line_end": 1534, "char_start": 81735, "char_end": 81759 }, "argument_gist": "", "excerpt": "v v_1 + w_1 \\vv w_1 \\in B_{\\vv r^+} \\left(2\\beta e^{s'}\\right)$. Then we define \\Delta(y_1, y_2) = (v_2, w_2 ). Let us call the collection of tubes $\\{\\cT(y): y \\in \\fJ\\}$ with weights $\\{\\cW(y)\\}$ the Kakeya-type model for $(\\tilde \\mu, \\Pi(x))$. \\par Now let us prove Lemma \\ref{lm:cross-section}: [Proof of Lemma \\ref{lm:cross-sec" }, { "id": "eq:z-z'", "kind": "anchor", "label": "eq:z-z'", "owner": "lm:partition-1", "section": "sec-kakeya-model", "anchor": { "source": "manuscript", "line_start": 1653, "line_end": 1654, "char_start": 90321, "char_end": 90336 }, "argument_gist": "", "excerpt": "$\\tilde b, \\tilde b_0 , \\tilde r^\\ast, \\tilde r \\in [0.1 \\beta e^{-2s'}]$. Thus, z' = a_0 a u^\\ast (\\bar r^\\ast) \\exp(\\vv w^-) \\exp(\\vv w^+) u(\\bar r) z, where $a_0 a \\in B^{A_0 A}(3.1 \\beta)$, $\\bar r^\\ast \\in [3.1 \\beta e^{-2s'}]$, $\\bar r \\in [3.1 \\beta]$, $\\vv w^- \\in B_{\\vv r^-}(2.3 \\beta e^{-s'})$, and $\\vv w^+ \\in B_{\\vv r^+}(2.3 \\beta e^{-s'})$. Let us co" }, { "id": "eq:z'-y", "kind": "anchor", "label": "eq:z'-y", "owner": "lm:partition-2", "section": "sec-kakeya-model", "anchor": { "source": "manuscript", "line_start": 1686, "line_end": 1687, "char_start": 92521, "char_end": 92536 }, "argument_gist": "", "excerpt": ".2 \\beta e^{-3s'}$. \\par We will prove that $z' \\in \\Omega(y)$. In fact, we have z' = a_0 a u(r) \\exp(\\vv w^+ + \\vv w^-) \\exp(-\\bar{\\vv w}^-) u^\\ast (r^\\ast)y. By Lemma \\ref{lm:product-w}, we have that \\[ \\exp(\\vv w^+ + \\vv w^-) \\exp(-\\bar{\\vv w}^-) = \\exp(\\vv w^+ + \\vv w^- - \\bar{\\vv w}^- + \\tilde{\\vv w} ) \\exp(\\tilde{\\vv h} ), \\] where $ \\tilde{\\vv w} \\in B_{" }, { "id": "eq:fC", "kind": "anchor", "label": "eq:fC", "owner": "rmk:partition-2", "section": "sec-kakeya-model", "anchor": { "source": "manuscript", "line_start": 1731, "line_end": 1732, "char_start": 95625, "char_end": 95638 }, "argument_gist": "", "excerpt": "{proof} \\par Let us denote \\fC := \\left[1.2 \\beta e^{-s'}\\right]^2. From Lemma \\ref{lm:partition-2}, we can see that for any $z' \\in \\Theta(z)$, \\[ \\fD_z(z') \\in \\fC. \\] \\par Suppose \\[ y_2 = \\exp(\\vv w^-) \\exp(\\vv w^+) u(r_0) y_1,\\] where $|r_0|" }, { "id": "eq:v2r", "kind": "anchor", "label": "eq:v2r", "owner": "lm:wv2r", "section": "sec-kakeya-model", "anchor": { "source": "manuscript", "line_start": 1747, "line_end": 1748, "char_start": 96426, "char_end": 96440 }, "argument_gist": "", "excerpt": "\\[ \\vv w^-(r) = v_2(r) \\vv v_2 + w_2 (r) \\vv w_2 \\in \\vv r^-. \\] Then v_2(r) = \\frac{2 v_2}{2 - r w_2 v_2}, and w_2(r) = \\frac{2 w_2}{2 + r w_2 v_2}. In particular, for any $r' \\in [\\beta e^{2s'}]$, if we denote $y'_1 = u(r') y_1$, $y'_2 = u(r) y_2$, we have \\[ \\|\\Delta(y'_1, y'" }, { "id": "eq:w2r", "kind": "anchor", "label": "eq:w2r", "owner": "lm:wv2r", "section": "sec-kakeya-model", "anchor": { "source": "manuscript", "line_start": 1752, "line_end": 1753, "char_start": 96531, "char_end": 96545 }, "argument_gist": "", "excerpt": "l{eq:v2r} v_2(r) = \\frac{2 v_2}{2 - r w_2 v_2}, and w_2(r) = \\frac{2 w_2}{2 + r w_2 v_2}. In particular, for any $r' \\in [\\beta e^{2s'}]$, if we denote $y'_1 = u(r') y_1$, $y'_2 = u(r) y_2$, we have \\[ \\|\\Delta(y'_1, y'_2)\\| \\asymp \\|\\Delta(y_1, y_2)\\|. \\] This can be verified by direct calculat" }, { "id": "eq:hat-omega", "kind": "anchor", "label": "eq:hat-omega", "owner": "lm:omega-u", "section": "sec-kakeya-model", "anchor": { "source": "manuscript", "line_start": 1767, "line_end": 1768, "char_start": 97121, "char_end": 97141 }, "argument_gist": "", "excerpt": "\\mu$ and $\\Pi(x)$ be as in Definition \\ref{def:kakeya-model}. Let us denote \\hat{\\Omega}_{s'} := Q^{s'}_{3s'}(2.1\\beta) Q^{A_0 H} (2s', 2.1\\beta). For $y \\in \\fJ$, let us denote U\\Omega'(y) : = \\left\\{ u(r') y' : y' \\in \\Omega'(y), r' \\in \\fl(y') \\right\\}. For $r \\in \\fl(y)$, let us denote \\begin{" }, { "id": "eq:u-omega", "kind": "anchor", "label": "eq:u-omega", "owner": "lm:omega-u", "section": "sec-kakeya-model", "anchor": { "source": "manuscript", "line_start": 1771, "line_end": 1772, "char_start": 97283, "char_end": 97301 }, "argument_gist": "", "excerpt": "eta) Q^{A_0 H} (2s', 2.1\\beta). For $y \\in \\fJ$, let us denote U\\Omega'(y) : = \\left\\{ u(r') y' : y' \\in \\Omega'(y), r' \\in \\fl(y') \\right\\}. For $r \\in \\fl(y)$, let us denote \\Omega(y, r) := U\\Omega'(y) \\cap \\Sigma^{\\tilde{r}}, where $\\tilde r$ satisfies that $u(r)y \\in \\Sigma^{\\tilde" }, { "id": "eq:omega-y-r", "kind": "anchor", "label": "eq:omega-y-r", "owner": "lm:omega-u", "section": "sec-kakeya-model", "anchor": { "source": "manuscript", "line_start": 1775, "line_end": 1776, "char_start": 97453, "char_end": 97473 }, "argument_gist": "", "excerpt": "y), r' \\in \\fl(y') \\right\\}. For $r \\in \\fl(y)$, let us denote \\Omega(y, r) := U\\Omega'(y) \\cap \\Sigma^{\\tilde{r}}, where $\\tilde r$ satisfies that $u(r)y \\in \\Sigma^{\\tilde{r}}$. Then \\[ \\Omega(y, r) \\subset \\hat{\\Omega}(u(r)y) := \\hat{\\Omega}_{s'} u(r) y. \\] In particular, we have \\[ \\tilde \\mu(\\Omega'(y)) \\ll \\tilde \\mu \\left(\\hat{\\Omeg" }, { "id": "eq:Iy1y2", "kind": "anchor", "label": "eq:Iy1y2", "owner": "lm:time-leave", "section": "sec-kakeya-model", "anchor": { "source": "manuscript", "line_start": 1817, "line_end": 1818, "char_start": 100198, "char_end": 100214 }, "argument_gist": "", "excerpt": "\\vv r^+.$$ Then $\\vv w^+ \\in B_{\\vv r^+}(4.6 \\beta e^{-s'})$. Moreover, let I (y_1, y_2) := \\left\\{ r_1 \\in \\fl(y_1) : \\exists r_2 , z' \\in \\cI, \\text{ s.t. } \\Omega (y_i, r_i) \\cap \\Theta'(z')\\neq \\emptyset, i=1,2 \\right\\}. Then there exists an interval $I$, such that $ I(y_1, y_2) \\subset I$ with $$|I| \\asymp \\min\\left\\{ \\beta e^{2s'}, \\beta e^{-s" }, { "id": "eq:hyperbola", "kind": "anchor", "label": "eq:hyperbola", "owner": "lm:concentrated-surface", "section": "sec-kakeya-model", "anchor": { "source": "manuscript", "line_start": 1866, "line_end": 1867, "char_start": 104010, "char_end": 104030 }, "argument_gist": "", "excerpt": "'}$ is contained in the $ \\beta^{-100} e^{-3s'}$ neighborhood of the hyperbola \\left\\{ (X, Y) \\in \\fC: (X - X_0)(Y-Y_0) + \\frac{X - X_0}{v_1} + \\frac{Y - Y_0}{w_1} = 0 \\right\\}, where $v_1, w_1, X_0, Y_0$ are constants depending on $y, z$ with bounds $(X_0, Y_0) \\in \\fC$ and $(v_1, w_1) \\in [\\beta e^{s'}]^2$. The set of such coordinates $\\fD_z(\\bar" }, { "id": "eq:hyperbola-box", "kind": "anchor", "label": "eq:hyperbola-box", "owner": "lm:concentrated-surface", "section": "sec-kakeya-model", "anchor": { "source": "manuscript", "line_start": 1872, "line_end": 1873, "char_start": 104385, "char_end": 104409 }, "argument_gist": "", "excerpt": "^2$. The set of such coordinates $\\fD_z(\\bar z)$ is also contained in the box \\fB(\\cT(y), z) := (X_0, Y_0) + \\left[\\beta^{-100} e^{-2s'} \\|(v_1, w_1)\\|\\right]^2. Moreover, if there exists $\\bar z \\in \\cI_z$ such that there is some $r \\in I(\\bar z, y)$ with $|r| \\ge \\beta^{100} e^{2s'}$ and \\[ \\|\\Delta(\\bar z , y)\\| \\le \\beta e^{-s'-j}, \\] then \\[ \\|" }, { "id": "eq:2", "kind": "anchor", "label": "eq:2", "owner": "lm:concentrated-surface", "section": "sec-kakeya-model", "anchor": { "source": "manuscript", "line_start": 1899, "line_end": 1900, "char_start": 105939, "char_end": 105951 }, "argument_gist": "", "excerpt": "w(r)| \\le 4.6 \\beta e^{-s'}$. Let us denote $\\Delta = v_2 w_2/2$. Then we have r^{-1} + \\Delta = - \\frac{v_2}{v_1} (1+ O(\\beta e^{-s'}v_1^{-1})), and r^{-1} - \\Delta = \\frac{w_2}{w_1} ( 1+ O(\\beta e^{-s'} w_1^{-1}) ). Note that $|v_1| \\ge \\beta e^{-s'}$. So we have \\[ |\\beta e^{-s'} v_1^{-1}| \\le 1.\\] Note that" }, { "id": "eq:1", "kind": "anchor", "label": "eq:1", "owner": "lm:concentrated-surface", "section": "sec-kakeya-model", "anchor": { "source": "manuscript", "line_start": 1903, "line_end": 1904, "char_start": 106059, "char_end": 106071 }, "argument_gist": "", "excerpt": "} + \\Delta = - \\frac{v_2}{v_1} (1+ O(\\beta e^{-s'}v_1^{-1})), and r^{-1} - \\Delta = \\frac{w_2}{w_1} ( 1+ O(\\beta e^{-s'} w_1^{-1}) ). Note that $|v_1| \\ge \\beta e^{-s'}$. So we have \\[ |\\beta e^{-s'} v_1^{-1}| \\le 1.\\] Note that |r^{-1}| \\ge \\beta^{-1} e^{-2s'} \\ge \\beta^2 e^{-2s'} \\ge |\\Delta|. Since" }, { "id": "eq:3", "kind": "anchor", "label": "eq:3", "owner": "lm:concentrated-surface", "section": "sec-kakeya-model", "anchor": { "source": "manuscript", "line_start": 1909, "line_end": 1910, "char_start": 106271, "char_end": 106283 }, "argument_gist": "", "excerpt": "| \\ge \\beta e^{-s'}$. So we have \\[ |\\beta e^{-s'} v_1^{-1}| \\le 1.\\] Note that |r^{-1}| \\ge \\beta^{-1} e^{-2s'} \\ge \\beta^2 e^{-2s'} \\ge |\\Delta|. Since $|r| \\ge \\beta^{100}e^{2s'}$, we get |v_2/v_1|, |w_2/ w_1| \\le \\beta^{-100}e^{-2s'}. This implies that $$\\|(v_2, w_2)\\| \\le \\beta^{-100}e^{-2s'}" }, { "id": "eq:control-w2", "kind": "anchor", "label": "eq:control-w2", "owner": "lm:concentrated-surface", "section": "sec-kakeya-model", "anchor": { "source": "manuscript", "line_start": 1913, "line_end": 1914, "char_start": 106429, "char_end": 106450 }, "argument_gist": "", "excerpt": "e^{-2s'} \\ge |\\Delta|. Since $|r| \\ge \\beta^{100}e^{2s'}$, we get |v_2/v_1|, |w_2/ w_1| \\le \\beta^{-100}e^{-2s'}. This implies that $$\\|(v_2, w_2)\\| \\le \\beta^{-100}e^{-2s'} \\|(v_1, w_1)\\| .$$ Moreover, \\eqref{eq:1} and \\eqref{eq:2} imply that \\[ v_2 w_2 + v_1^{-1} v_2 + w_1^{-1} w_2 = O(\\beta^{-99} e^{-3s'} v_1^{-1}) + O(\\beta^{-99}" }, { "id": "item:line", "kind": "anchor", "label": "item:line", "owner": "rmk:concentrated-surface", "section": "sec-kakeya-model", "anchor": { "source": "manuscript", "line_start": 1941, "line_end": 1941, "char_start": 108280, "char_end": 108297 }, "argument_gist": "", "excerpt": "1, y_2) \\] such that $|P_{y_1, y'_1} (r') - r'| \\le \\beta |r'|$ for every $r' \\in I(y_1,y_2)$. \\item If $|v_1/w_1| \\le e^{-s'}$ or $|w_1/v_1| \\le e^{-s'}$, then the neighborhood described in~\\eqref{eq:hyperbola} agrees with the $ \\beta^{-100} e^{-3s'}$ neighborhood of the straight line \\[ v_1^{-1}(X - X_0) + w_1^{-1}(Y - Y_0) = 0.\\] \\medskip \\par Let us denot" }, { "id": "eq:heavy-part-is-small", "kind": "anchor", "label": "eq:heavy-part-is-small", "owner": "sec-dimension-improvement-unstructured", "section": "sec-dimension-improvement-unstructured", "anchor": { "source": "manuscript", "line_start": 1990, "line_end": 1991, "char_start": 111064, "char_end": 111094 }, "argument_gist": "", "excerpt": "r To prove Proposition \\ref{prop:improve-dimension}, it suffices to show that \\tilde \\mu(\\cR_i) \\le \\beta^{20}. In fact, if this is the case, we can put $\\tilde \\mu_{\\cR_i} = \\tilde{\\mu}^{-1}(\\cR_i)\\tilde{\\mu}|_{ \\cR_i}$, and $\\tilde \\mu_{i+1}= \\tilde \\mu^{-1}(\\Pi(x)\\setminus\\cR_i)\\tilde \\mu_{\\fr}|_{\\Pi(x)\\setminus\\cR_i}$. Then since $\\Pi(x)\\setminus\\cR_" }, { "id": "eq:upper-omega", "kind": "anchor", "label": "eq:upper-omega", "owner": "lm:omega-upper", "section": "sec-dimension-improvement-unstructured", "anchor": { "source": "manuscript", "line_start": 2003, "line_end": 2004, "char_start": 112033, "char_end": 112055 }, "argument_gist": "", "excerpt": "Let us denote $s' = s_{i+1}$. For any $\\Omega(y) = \\Omega_{s'} y$, we have \\tilde \\mu (\\Omega'(y)) \\le \\beta^{-15}e^{-2 s'} e^{-2 d_i s'}. First note that \\[ \\mu(\\Omega(y)) = \\mu_{\\cF_i}(a(-2s') \\Omega(y)),\\] and \\[ a(-2s') \\Omega(y) = Q_{2s'}(2\\beta) R^{A_0 H}(2s', 2\\beta) y' \\] where $y' = a(-2s') y$. Noting that $\\mu_{\\c" }, { "id": "eq:strange-normal-decomposition", "kind": "anchor", "label": "eq:strange-normal-decomposition", "owner": "def:heavy-part-decomposition", "section": "sec-dimension-improvement-unstructured", "anchor": { "source": "manuscript", "line_start": 2030, "line_end": 2031, "char_start": 113569, "char_end": 113608 }, "argument_gist": "", "excerpt": "neighborhoods that are also normal. Then we have the following decomposition: \\cR_i = \\bar \\cR_i \\cup \\bigcup_{j = C_1 \\epsilon_1 s'}^{2s'} \\cR_{i,j}, where $\\cR_{i,j}$ denotes the union of heavy neighborhoods $\\Theta'(z)$ such that there exists $\\Xi_j(z') \\subset \\Theta\u2019(z)$ satisfying \\tilde \\mu_{\\fr}(" }, { "id": "eq:strange-nbhd-condition", "kind": "anchor", "label": "eq:strange-nbhd-condition", "owner": "def:heavy-part-decomposition", "section": "sec-dimension-improvement-unstructured", "anchor": { "source": "manuscript", "line_start": 2035, "line_end": 2036, "char_start": 113857, "char_end": 113890 }, "argument_gist": "", "excerpt": "ds $\\Theta'(z)$ such that there exists $\\Xi_j(z') \\subset \\Theta\u2019(z)$ satisfying \\tilde \\mu_{\\fr}(\\Xi_j(z')) \\ge (1-\\beta)^{j} \\tilde \\mu_{\\fr}(\\Theta'(z)), and \\tilde \\mu_{\\fr}(\\Xi_{j+1}(z'')) < (1-\\beta) \\tilde \\mu_{\\fr}(\\Xi_j(z')) for any $\\Xi_{j+1}(z'') \\subset \\Xi_j(z')$. Here $j \\le 2s'$" }, { "id": "eq:strange-nbhd-condition-2", "kind": "anchor", "label": "eq:strange-nbhd-condition-2", "owner": "def:heavy-part-decomposition", "section": "sec-dimension-improvement-unstructured", "anchor": { "source": "manuscript", "line_start": 2039, "line_end": 2040, "char_start": 114005, "char_end": 114040 }, "argument_gist": "", "excerpt": "r}(\\Xi_j(z')) \\ge (1-\\beta)^{j} \\tilde \\mu_{\\fr}(\\Theta'(z)), and \\tilde \\mu_{\\fr}(\\Xi_{j+1}(z'')) < (1-\\beta) \\tilde \\mu_{\\fr}(\\Xi_j(z')) for any $\\Xi_{j+1}(z'') \\subset \\Xi_j(z')$. Here $j \\le 2s'$ because for $j > 2s'$, \\[ \\Xi_j (z') \\subset \\Omega(z'), \\] and by \\eqref{eq:upper-omega}, \\[ \\tilde \\mu (\\Omega(z')) \\le \\beta^{-15}e^{-2 s'} e^{-2" }, { "id": "eq:fScT", "kind": "anchor", "label": "eq:fScT", "owner": "def:structure-randomness", "section": "sec-dimension-improvement-unstructured", "anchor": { "source": "manuscript", "line_start": 2059, "line_end": 2060, "char_start": 115518, "char_end": 115533 }, "argument_gist": "", "excerpt": "n it is easy to see that \\fS(\\cT(y), z) \\le N_2 \\beta^{-100} \\beta^{-100}e^{2s' - j} e^{-2 d_i s' } e^{-2s'} \\le \\beta^{-201} e^{-2 d_i s' - j}, where $N_2$ is a large numeric constant (in \\S\\ref{subsec-constants},it was chosen to be $10^6$). The curve neighborhood $\\cC(\\cT(y), z)$ is called \\emph{" }, { "id": "eq:structure-part-theta-0", "kind": "anchor", "label": "eq:structure-part-theta-0", "owner": "def:structure-random-decompostion", "section": "sec-dimension-improvement-unstructured", "anchor": { "source": "manuscript", "line_start": 2082, "line_end": 2083, "char_start": 116725, "char_end": 116758 }, "argument_gist": "", "excerpt": "is heavy and normal, and $\\Sigma^r$ such that $z \\in \\Sigma^r$, let us define: \\HS(z,r) := \\left\\{ y \\in \\fJ\\cap \\Sigma^r: \\cC(\\cT(y), z) \\subset \\fC \\text{ is highly concentrated} \\right\\}, and \\SS(z,r) := \\bigcup_{y \\in \\HS(z, r)} \\Omega'(y). If there exists $\\Sigma^r$ c" }, { "id": "eq:structure-set-theta-0", "kind": "anchor", "label": "eq:structure-set-theta-0", "owner": "def:structure-random-decompostion", "section": "sec-dimension-improvement-unstructured", "anchor": { "source": "manuscript", "line_start": 2087, "line_end": 2088, "char_start": 116923, "char_end": 116955 }, "argument_gist": "", "excerpt": "ubset \\fC \\text{ is highly concentrated} \\right\\}, and \\SS(z,r) := \\bigcup_{y \\in \\HS(z, r)} \\Omega'(y). If there exists $\\Sigma^r$ containing $z$ such that \\[ \\tilde \\mu (\\SS(z,r)) \\ge e^{-2s' - 10 C_1 \\epsilon_1 s'}, \\] we will call $\\Theta'(z)$ highly structured. Then we have the following decomposition for $\\bar \\" }, { "id": "eq:normal-random-structure-decomposition", "kind": "anchor", "label": "eq:normal-random-structure-decomposition", "owner": "def:structure-random-decompostion", "section": "sec-dimension-improvement-unstructured", "anchor": { "source": "manuscript", "line_start": 2094, "line_end": 2095, "char_start": 117280, "char_end": 117328 }, "argument_gist": "", "excerpt": "ph{highly structured}. Then we have the following decomposition for $\\bar \\cR_i$: \\bar \\cR_i = \\bar \\cR^{\\fr}_{i} \\cup \\bar \\cR^{\\fs}_{i}, where $\\bar \\cR^{\\fs}_i$ denotes the union of heavy, normal, and highly structured $\\Theta'(z)$, and $\\bar \\cR^{\\fr}_i$ denotes the union of heavy, normal $\\Theta'(z)$ which are not highly structured. \\par We first pr" }, { "id": "eq:assumption-1", "kind": "anchor", "label": "eq:assumption-1", "owner": "prop:dimension-improvement-random", "section": "sec-dimension-improvement-unstructured", "anchor": { "source": "manuscript", "line_start": 2149, "line_end": 2150, "char_start": 121110, "char_end": 121133 }, "argument_gist": "", "excerpt": "hen $\\tilde z \\in \\cI_{z;y'}$ implies that for any $\\hat z \\in \\fJ(\\tilde z, r)$, \\| \\Delta(y, \\hat z) \\| \\gtrsim \\beta e^{-(1 + C_1 \\epsilon_1 ) s'}. \\par Noting that $\\tilde \\mu_{\\fr}(\\Theta'(z)" }, { "id": "eq:cS-lower-bound", "kind": "anchor", "label": "eq:cS-lower-bound", "owner": "prop:dimension-improvement-random", "section": "sec-dimension-improvement-unstructured", "anchor": { "source": "manuscript", "line_start": 2173, "line_end": 2174, "char_start": 123362, "char_end": 123387 }, "argument_gist": "", "excerpt": "\\in \\fJ \\cap \\Sigma^r$ such that $\\Omega'(y) \\cap \\cR'_i \\neq \\emptyset$, we get \\cS(\\Sigma^r) \\ge \\beta^{62} \\tilde{\\fm} e^{- (d_i + \\epsilon_1) s' - C_1 \\epsilon_1 s'} \\ge \\tilde{\\fm} e^{-d_i s' -2 C_1 \\epsilon_1 s'}, where $\\tilde{\\fm} = \\tilde \\mu_{\\fr}(\\Sigma^r) \\asymp e^{-2s'}$. \\par On the other hand, let us get an upper bound on $\\cS(\\Sigma^r)$. \\par For" }, { "id": "eq: triple sum", "kind": "anchor", "label": "eq: triple sum", "owner": "prop:dimension-improvement-strange-part", "section": "sec-dimension-improvement-unstructured", "anchor": { "source": "manuscript", "line_start": 2268, "line_end": 2268, "char_start": 130656, "char_end": 130678 }, "argument_gist": "", "excerpt": "um_{y' \\in \\cY_j} \\, \\sum_{y'_2 \\in \\cI_{y'\\!, j} } \\,\\sum_{y_2 \\in \\fJ_j(y'_2, r)} \\cW(y) \\cW(y_2) \\\\ &\\qquad \\gg \\sum_{y' \\in \\cY_j} \\cW(y)e^{-2s'} e^{- (d_i+2\\epsilon_1)s' } \\nonumber\\\\ &\\qquad \\gg \\beta^{29} e^{2s'} \\cW(y)e^{-2s'} e^{- (d_i+2\\epsilon_1)s' } = \\beta^{29} \\cW(y) e^{- (d_i+2\\epsilon_1)s' }.\\nonumber Since $\\|\\Delta(y, y_2)\\| \\ge \\beta e^{-s' - j-1}$, by Lemma \\" }, { "id": "eq:j lower bound", "kind": "anchor", "label": "eq:j lower bound", "owner": "prop:dimension-improvement-strange-part", "section": "sec-dimension-improvement-unstructured", "anchor": { "source": "manuscript", "line_start": 2280, "line_end": 2281, "char_start": 131857, "char_end": 131881 }, "argument_gist": "", "excerpt": "\\in \\fJ \\cap \\Sigma^r$ with $\\Omega'(y) \\cap \\cR''_{i,j} \\neq \\emptyset$, we have \\cS_j(\\Sigma^r) \\ge \\tilde \\mu(\\cR''_{i,j}\\cap \\Sigma^r) e^{- (d_i + 3 \\epsilon_1)s' - j} \\ge \\beta^{30} \\tilde \\fm e^{- (d_i + 3 \\epsilon_1)s' - j}, where $\\tilde \\fm = \\tilde \\mu(\\Sigma^r)$. \\medskip \\par Now let us get an upper bound on $\\cS_j(\\Sigma^r)$. To do this, we need th" }, { "id": "eq:hat-y1-y2", "kind": "anchor", "label": "eq:hat-y1-y2", "owner": "lem:sigma j lemma", "section": "sec-dimension-improvement-unstructured", "anchor": { "source": "manuscript", "line_start": 2310, "line_end": 2311, "char_start": 133893, "char_end": 133913 }, "argument_gist": "", "excerpt": "{-j})$, and $\\hat{\\vv h} \\in B_{\\R \\fa_0 + \\fh} (0.1 \\beta e^{-2j})$. Therefore, \\hat y_1 = \\exp(\\tilde{\\vv w}) \\exp(\\tilde{\\vv h}) \\hat y_2, where $\\tilde{\\vv w} \\in B_{\\fr_1 + \\fr_2} (1.3 \\beta e^{-j})$ and $\\tilde{\\vv h} \\in B_{\\R \\fa_0 + \\fh}(1.1\\beta)$. Note that $y_1 \\in \\Xi_j (z')$, we have \\[ y_1 = \\exp(\\vv w^+_1 + \\vv w^-_1) \\exp(\\vv h_1) z', \\] where" }, { "id": "eq:hat-y1-z", "kind": "anchor", "label": "eq:hat-y1-z", "owner": "lem:sigma j lemma", "section": "sec-dimension-improvement-unstructured", "anchor": { "source": "manuscript", "line_start": 2317, "line_end": 2318, "char_start": 134419, "char_end": 134438 }, "argument_gist": "", "excerpt": "\\in \\fq^{A_0 H}(2s', \\beta)$. Let us denote $\\hat z' = a(-2s') z'$, then we have \\hat y_1 = \\exp(\\tilde{\\vv w}_1) \\exp(\\tilde{\\vv h}_1) \\hat z', where $\\tilde{\\vv w}_1 \\in B_{\\fr_1 + \\fr_2}(\\beta e^{-j})$, and $\\tilde{\\vv h}_1 \\in B_{\\R \\fa_0 + \\fh}(\\beta)$. \\eqref{eq:hat-y1-y2} and \\eqref{eq:hat-y1-z} imply that \\[ \\hat y_2 = \\exp(-\\tilde{\\vv h}) \\exp(-\\tilde{\\" }, { "id": "eq:cIy2j", "kind": "anchor", "label": "eq:cIy2j", "owner": "prop:dimension-improvement-strange-part", "section": "sec-dimension-improvement-unstructured", "anchor": { "source": "manuscript", "line_start": 2333, "line_end": 2334, "char_start": 135602, "char_end": 135618 }, "argument_gist": "", "excerpt": "j lemma}, we can prove that \\[ \\Xi_j(z') \\subset \\Sigma_{s', j-2}(y_2). \\] Define \\cI(y_2, j) := \\Bigl\\{ z \\in \\cI \\cap \\Sigma^r \\cap \\cR_{i,j}\\!: &\\,\\text{the heavy slice $\\Xi_j(z') \\subset \\Theta'(z)$} \\\\ &\\text{ satisfies } \\Xi_j(z') \\subset \\Sigma_{s', j-2}(y_2) \\cap \\cR'_{i,j} \\Bigr\\}. Fix $z \\in \\cI(y_2, j)$. By Lemma \\ref{lm:co" }, { "id": "eq:upper-theta", "kind": "anchor", "label": "eq:upper-theta", "owner": "prop:dimension-does-not-drop", "section": "sec-no-loss-of-dimension", "anchor": { "source": "manuscript", "line_start": 2386, "line_end": 2387, "char_start": 139328, "char_end": 139350 }, "argument_gist": "", "excerpt": "u(\\mathfrak Z) \\le \\beta^{50}$ such that for $x \\in \\cI\\setminus\\mathfrak{Z}$, \\tilde \\mu(\\Theta'(x)) \\le e^{-2s'} e^{-(d_i - C_1 \\epsilon_1)s'}. [Sketch of the proof] \\par For $z \\in \\cI$, let us call $\\Theta'(z)$ extremely heavy if \\[ \\tilde \\mu(\\Theta'(z)) > \\tilde M_{s'} \\qquad\\text{for $ \\tilde M_{s'}= e^{-2s'} e^{" }, { "id": "eq: y_2 sum for diamond'", "kind": "anchor", "label": "eq: y_2 sum for diamond'", "owner": "prop:dimension-does-not-drop", "section": "sec-no-loss-of-dimension", "anchor": { "source": "manuscript", "line_start": 2432, "line_end": 2433, "char_start": 142073, "char_end": 142105 }, "argument_gist": "", "excerpt": "\\cap \\Sigma^r$ such that \\[ \\Omega'(y_1) \\cap \\fZ'' \\neq \\emptyset, \\] the summation \\sum_{y_2: (y_1, y_2) \\in \\diamond'_r} \\cW(y_1) \\cW(y_2) &\\gg \\beta^{52} e^{2s' - C_1 \\epsilon_1 s'} \\beta \\tilde{M}_{s'} \\cW(y_1) , since the trajectory $u(\\fl(y)) y \\cap \\fZ'$ encounters $\\gtrapprox \\beta ^{51} e^{2s'}$ many very heavy $\\Theta'$ neighbourhoods sufficiently far fro" }, { "id": "eq:theta sum'", "kind": "anchor", "label": "eq:theta sum'", "owner": "prop:dimension-does-not-drop", "section": "sec-no-loss-of-dimension", "anchor": { "source": "manuscript", "line_start": 2436, "line_end": 2437, "char_start": 142464, "char_end": 142485 }, "argument_gist": "", "excerpt": "eta'$ neighbourhoods sufficiently far from $\\Sigma^r$. For each such $\\Theta'(z)$, \\tilde M_{s'} \\leq \\tilde \\mu (\\Theta'(z)) \\approx \\sum_{y_2}\\!{\\vphantom{\\sum}}' \\cW(y_2) where the sum is over all $y_2 \\in \\fJ \\cap \\Sigma^r$ so that $(y_1,y_2) \\in \\diamond_r'$ and there is a $|r'|\\leq \\beta e^{2s'}$ and $y_2' \\in \\Omega(y_2)$ so $u(r') y_2' \\in \\Theta'(z)$. Howe" }, { "id": "eq:full-omega", "kind": "anchor", "label": "eq:full-omega", "owner": "def:full-omega", "section": "sec-structured-component", "anchor": { "source": "manuscript", "line_start": 2576, "line_end": 2577, "char_start": 151330, "char_end": 151351 }, "argument_gist": "", "excerpt": "irst fix some notation. For $z \\in \\fJ$, if \\tilde \\mu(\\Omega'(z)) \\ge e^{-2s'} e^{-2d_i s' - 8 C_1 \\epsilon_1 s'}, we will call $\\Omega'(z)$ full. Otherwise, we call it almost empty. For $z \\in \\cI$, we say that two curve neighborhoods $\\cC = \\cC(\\cT(y), z)$ a" }, { "id": "eq:w1w'1", "kind": "anchor", "label": "eq:w1w'1", "owner": "lm:hyperbola-plane", "section": "sec-structured-component", "anchor": { "source": "manuscript", "line_start": 2596, "line_end": 2597, "char_start": 152605, "char_end": 152621 }, "argument_gist": "", "excerpt": "}$, if $C'$ is contained in the $A e^{-3s'}$ neighborhood of $C$, then we have |w_{1} - w'_{1}| \\le Ae^{-s'+2\\ell}, \\text{ and } |k - k'| \\le Ae^{-2s' +2\\ell}. Conversely, if \\eqref{eq:w1w'1} holds, then $C'$ is contained in the $A e^{-3s'}$ neighborhood of $C$. \\par (2) Suppose $C'$ is not contained in the $e^{-3s' + 11 C_1 \\epsilon_1 s'}$ nei" }, { "id": "eq:w1-w1'", "kind": "anchor", "label": "eq:w1-w1'", "owner": "lm:hyperbola-plane", "section": "sec-structured-component", "anchor": { "source": "manuscript", "line_start": 2601, "line_end": 2602, "char_start": 153010, "char_end": 153027 }, "argument_gist": "", "excerpt": "_1 s'}$ neighborhood of $C$. For $0< A < e^{2s'-C_1\\epsilon_1 s'}$, if we have |w_{1} - w'_{1}| \\in \\left[Ae^{-s'+2\\ell} , Ae^{-s'+2\\ell+1}\\right], and there exists $ \\vv p' \\in C'$ such that $\\|\\vv p'\\| \\ge e^{-s' -10 C_1 \\epsilon_1 s'}$ and $\\vv p'$ is contained in the $\\beta^{-100} e^{-3s'}$ neighborhood of $C$, then $C'$ is contained in the $" }, { "id": "eq:y-y1-lhs", "kind": "anchor", "label": "eq:y-y1-lhs", "owner": "sec-structured-component", "section": "sec-structured-component", "anchor": { "source": "manuscript", "line_start": 2654, "line_end": 2655, "char_start": 156015, "char_end": 156034 }, "argument_gist": "", "excerpt": "+ k'|} \\le Ae^{-3s'}. \\] Note that the left-hand side is bounded from above by e^{-s'- 2\\ell} |w_{1} - w'_{1}| \\cdot \\left| Y + \\frac{k' - k}{w_{1} - w'_{1}} \\right|. Let us consider the 2 subcases given as above. \\par For Subcase 1, we have |w_{1} - w'_{1}| \\cdot \\left| Y + \\frac{k' - k}{w_{1} - w'_{1}} \\right| &\\le e^{-s'}|w" }, { "id": "eq:k over w claim", "kind": "anchor", "label": "eq:k over w claim", "owner": "sec-structured-component", "section": "sec-structured-component", "anchor": { "source": "manuscript", "line_start": 2673, "line_end": 2674, "char_start": 157349, "char_end": 157374 }, "argument_gist": "", "excerpt": "is easy to see that $|k'| \\in [e^{\\ell -1}, e^{\\ell +1}]$. \\par We claim that \\left| \\frac{k' - k}{w'_{1} - w_1} \\right| \\le e^{-s'}. In fact, if this is not the case, we have, \\[ \\left| \\frac{k' - k}{w'_{1} - w_1} \\right| \\ll \\left| \\tilde Y + \\frac{k' - k}{w'_{1} - w_1} \\right|, \\] which implies that \\[ \\frac{| \\tilde Y|\\cdot |k'-k|}{|w_" }, { "id": "eq:prime curve", "kind": "anchor", "label": "eq:prime curve", "owner": "lm:integrable-surface", "section": "sec-structured-component", "anchor": { "source": "manuscript", "line_start": 2737, "line_end": 2738, "char_start": 161573, "char_end": 161595 }, "argument_gist": "", "excerpt": "1, y'} \\vv v_1 + w_{1, y'} \\vv w_1) u(r_{y'}) y_0', \\] we get that the curve XY + v_{1, y'}^{-1} X + w_{1, y'}^{-1} Y = 0 is contained in $ 2e^{-3s' + 10 C_1 \\epsilon_1 s'}$ neighborhood of the curve XY + v_{1,y}^{-1} X + w_{1,y}^{-1} Y = 0. Let us assume that $|w_{1,y}| \\ge |v_{1,y}|$. T" }, { "id": "eq:not prime curve", "kind": "anchor", "label": "eq:not prime curve", "owner": "lm:integrable-surface", "section": "sec-structured-component", "anchor": { "source": "manuscript", "line_start": 2742, "line_end": 2742, "char_start": 161761, "char_end": 161787 }, "argument_gist": "", "excerpt": "is contained in $ 2e^{-3s' + 10 C_1 \\epsilon_1 s'}$ neighborhood of the curve XY + v_{1,y}^{-1} X + w_{1,y}^{-1} Y = 0. Let us assume that $|w_{1,y}| \\ge |v_{1,y}|$. The opposite case can be handled similarly. \\par Let us denote \\[ k = \\frac{w_{1,y}}{v_{1,y}}, \\text{ and } k' = \\frac{w_{1, y'}}{v_{1, y'}}. \\] Then it is easy to see that $k \\in [e^\\e" }, { "id": "eq:counting-conditions", "kind": "anchor", "label": "eq:counting-conditions", "owner": "lm:counting-curves-nbhd", "section": "sec-structured-component", "anchor": { "source": "manuscript", "line_start": 2834, "line_end": 2835, "char_start": 168079, "char_end": 168109 }, "argument_gist": "", "excerpt": "\\] By Lemma \\ref{lm:hyperbola-plane} (1) with $A = e^{2s' - m - 2 \\ell}$, we have |w_1 - w'_{1}| \\le e^{s' -m }, \\text{ and } |k - k'| \\le e^{-m }. Note that by Lemma \\ref{lm:hyperbola-plane} (1) with $A = e^{11C_1 \\epsilon_1 s'}$, if $$|w'_1 - \\bar w'_1| \\le e^{-s'+ 11C_1 \\epsilon_1 s'+2\\ell}$$ and $$|k' - \\bar k'| \\le e^{-2s'+ 11C_1 \\epsilon_1 s'+2\\ell}," }, { "id": "eq:tildey-tildez", "kind": "anchor", "label": "eq:tildey-tildez", "owner": "sec-structured-component", "section": "sec-structured-component", "anchor": { "source": "manuscript", "line_start": 2878, "line_end": 2879, "char_start": 170277, "char_end": 170301 }, "argument_gist": "", "excerpt": "here are $\\tilde y \\in \\Omega(y)$ and $\\tilde z \\in \\Theta(z)$ such that \\tilde y =u(r) \\exp(\\vv w^+) \\tilde z, and \\[ \\tilde z = \\exp(\\vv w^-) z \\] where $\\vv w^+ = v_1 \\vv v_1 + w_1 \\vv w_1 \\in B_{\\vv r^+}(\\beta e^{s'})$, $r \\in [\\beta e^{2s'}]$, and $$\\vv w^- = X_1 \\vv v_2 + Y_1 \\vv w_2 \\in \\vv{r}^-.$$ By Lemma \\ref{lm:conce" }, { "id": "eq:pv1", "kind": "anchor", "label": "eq:pv1", "owner": "sec-structured-component", "section": "sec-structured-component", "anchor": { "source": "manuscript", "line_start": 2897, "line_end": 2898, "char_start": 171480, "char_end": 171494 }, "argument_gist": "", "excerpt": "\\beta^2$, hence \\[ |k| = |w_1/v_1| \\ge e^{\\ell}. \\] Therefore, we have |v_1| \\le e^{-\\ell} |w_1| \\le \\beta e^{s' - \\ell}. \\par Let $z' \\in \\Theta(z)$ be the following element: \\[ z' = \\exp(\\vv w_0^-) z,\\] where $\\vv w_0^- = X_0 \\vv v_2 + Y_0 \\vv w_2$. Then we have that \\tilde z = \\" }, { "id": "eq:tildez-z'", "kind": "anchor", "label": "eq:tildez-z'", "owner": "sec-structured-component", "section": "sec-structured-component", "anchor": { "source": "manuscript", "line_start": 2904, "line_end": 2905, "char_start": 171756, "char_end": 171776 }, "argument_gist": "", "excerpt": "z,\\] where $\\vv w_0^- = X_0 \\vv v_2 + Y_0 \\vv w_2$. Then we have that \\tilde z = \\exp(\\tilde{\\vv w}^-) u^\\ast (r^\\ast)z' where $|r^\\ast| \\le \\beta e^{-2s'}$ and $\\tilde{\\vv w}^- = \\tilde X \\vv v_2 + \\tilde Y \\vv w_2 $ with $(\\tilde X, \\tilde Y) \\in \\cN^\\ell (\\vv 0)$. \\par By \\eqref{eq:tildey-tildez} and \\eqref{eq:tildez-z'}, we get \\" }, { "id": "eq:upper-theta-2", "kind": "anchor", "label": "eq:upper-theta-2", "owner": "prop:dimension-improvement-structured-component", "section": "sec-structured-component", "anchor": { "source": "manuscript", "line_start": 2949, "line_end": 2950, "char_start": 174801, "char_end": 174825 }, "argument_gist": "", "excerpt": "C_1 \\epsilon_1) s'} \\qquad \\text{for any $z \\in \\cI \\cap \\bar \\cR^{\\fs}_i$} { \\par Let us fix a $z \\in \\cI\\cap \\bar \\cR^{\\fs}_i$. By the definition of $\\bar \\cR^{\\fs}_i$ (see Definition \\ref{def:structure-random-decompostion}), we have that $\\Theta'(z)$ is highly structured. Thus, there exists $r=r(z)$ such that $z \\in \\Sigma^r$ and \\begin{equati" }, { "id": "eq:SSzr", "kind": "anchor", "label": "eq:SSzr", "owner": "prop:dimension-improvement-structured-component", "section": "sec-structured-component", "anchor": { "source": "manuscript", "line_start": 2954, "line_end": 2955, "char_start": 175128, "char_end": 175143 }, "argument_gist": "", "excerpt": "is highly structured. Thus, there exists $r=r(z)$ such that $z \\in \\Sigma^r$ and \\tilde \\mu (\\SS(z,r)) \\ge e^{-2s' - 10 C_1 \\epsilon_1 s'} Recall that by \\eqref{eq:upper-omega}, for any $y$, \\[ \\tilde \\mu(\\Omega'(y)) \\le \\beta^{-15}e^{-2 s'} e^{-2 d_i s'}. \\] Recall also that $\\SS(z,r)$ is the union of $\\Omega'(y)$ with $y \\in \\HS(z,r)$, i.e" }, { "id": "eq:HSunion", "kind": "anchor", "label": "eq:HSunion", "owner": "prop:dimension-improvement-structured-component", "section": "sec-structured-component", "anchor": { "source": "manuscript", "line_start": 2970, "line_end": 2971, "char_start": 176239, "char_end": 176257 }, "argument_gist": "", "excerpt": "qref{eq:upper-theta-2}, it is enough to show that if $\\clubsuit$ holds, the union \\bigcup_{y \\in \\HS(z,r)}\\cC(\\cT(y),z) (which by definition corresponds to a subset of $\\Theta'(z)$) must cover at least $e^{d_i s' + 20 C_1 \\epsilon_1 s'}$ different full $\\Omega'(z')$ with $z' \\in\\cI_z$. Indeed, if this holds, then by the definition of full neighborhoods" }, { "id": "eq:nowweaim", "kind": "anchor", "label": "eq:nowweaim", "owner": "prop:dimension-improvement-structured-component", "section": "sec-structured-component", "anchor": { "source": "manuscript", "line_start": 2981, "line_end": 2982, "char_start": 176821, "char_end": 176840 }, "argument_gist": "", "excerpt": ". \\medskip \\par Now we aim to show that under the assumption $\\clubsuit$, \\bigcup_{y \\in \\HS(z,r)}\\!\\!\\!\\cC(\\cT(y),z)\\ \\ \\text{covers $\\geq e^{d_i s' + 20 C_1 \\epsilon_1 s'}$ different full $\\Omega'(z')$, $z' \\in \\cI_z$.} Given a highly concentrated $\\cC$ with equation \\[ (X- X_0) (Y- Y_0) + v_1^{-1}(X- X_0) + w_1^{-1}(Y- Y_0) = 0, \\]" }, { "id": "eq:def-hszrn", "kind": "anchor", "label": "eq:def-hszrn", "owner": "prop:dimension-improvement-structured-component", "section": "sec-structured-component", "anchor": { "source": "manuscript", "line_start": 3030, "line_end": 3031, "char_start": 180776, "char_end": 180796 }, "argument_gist": "", "excerpt": "ructure-part-theta-0} and \\eqref{eq:structure-set-theta-0}), let us define \\HS(z, r; \\cN^{s'}_1) := \\left\\{ y \\in \\fJ\\cap \\Sigma^r: \\cC(\\cT(y), z) \\subset \\cN^{s'}_1 \\ \\text{and is highly concentrated}\\right\\}, and \\SS(z, r; \\cN^{s'}_1) := \\bigcup_{y \\in \\HS(z, r; \\cN^{s'}_1 )} \\Omega'(y). W" }, { "id": "eq:def-sszrn", "kind": "anchor", "label": "eq:def-sszrn", "owner": "prop:dimension-improvement-structured-component", "section": "sec-structured-component", "anchor": { "source": "manuscript", "line_start": 3034, "line_end": 3035, "char_start": 180979, "char_end": 180999 }, "argument_gist": "", "excerpt": "\\cN^{s'}_1 \\ \\text{and is highly concentrated}\\right\\}, and \\SS(z, r; \\cN^{s'}_1) := \\bigcup_{y \\in \\HS(z, r; \\cN^{s'}_1 )} \\Omega'(y). We claim that \\tilde \\mu(\\SS(z,r; \\cN^{s'}_{1})) \\le e^{-2s'}e^{-800 C_1 \\epsilon_1 s'}. Indeed, by Lemma \\ref{lm:curve-in-N}, there exist" }, { "id": "eq:sszrN-small", "kind": "anchor", "label": "eq:sszrN-small", "owner": "prop:dimension-improvement-structured-component", "section": "sec-structured-component", "anchor": { "source": "manuscript", "line_start": 3038, "line_end": 3039, "char_start": 181132, "char_end": 181154 }, "argument_gist": "", "excerpt": "up_{y \\in \\HS(z, r; \\cN^{s'}_1 )} \\Omega'(y). We claim that \\tilde \\mu(\\SS(z,r; \\cN^{s'}_{1})) \\le e^{-2s'}e^{-800 C_1 \\epsilon_1 s'}. Indeed, by Lemma \\ref{lm:curve-in-N}, there exists $z' \\in \\Theta'(z)$ so that the following holds: for any $y \\in \\HS(z, r; \\cN_1^{s'} )$, we have \\[ \\Omega'(y) \\subset a(2s')Q(\\fr_1, s'-1, \\beta ) B" }, { "id": "eq:Lnm", "kind": "anchor", "label": "eq:Lnm", "owner": "prop:dimension-improvement-structured-component", "section": "sec-structured-component", "anchor": { "source": "manuscript", "line_start": 3113, "line_end": 3114, "char_start": 186659, "char_end": 186673 }, "argument_gist": "", "excerpt": "- Y_0) + {v'_1}^{-1} (X - X_0) + {w'_1}^{-1} (Y - Y_0) = 0, \\] satisfies that |w_1 - w'_1| \\in [e^{s' - m_{\\cC}}, e^{s' - m_{\\cC}+1}]. We take $m_{\\cC}$ to be the minimal natural number satisfying this condition. Let us denote the collection of the curve neighborhoods satisfying \\eqref{eq:Lnm} by $\\mathfrak{L}_n(\\cC, m_{\\cC})$. We also remove $\\cC" }, { "id": "eq:curve neighborhoods cover", "kind": "anchor", "label": "eq:curve neighborhoods cover", "owner": "prop:dimension-improvement-structured-component", "section": "sec-structured-component", "anchor": { "source": "manuscript", "line_start": 3131, "line_end": 3132, "char_start": 188256, "char_end": 188292 }, "argument_gist": "", "excerpt": "Thus, curve neighborhoods from $\\mathfrak{L}_n(\\cC, m_{\\cC})$ cover at least e^{4s' - m_{\\cC} - 2 \\ell - 31 C_1 \\epsilon_1 s'} different full $\\Omega'(z')$ in the $(\\ell, m_{\\cC})$-neighborhood of $\\cC$. We say that a $(\\ell, m')$-neighborhood is $\\eta$-almost filled if it covers at least $\\eta e^{4s' -" }, { "id": "eq:fN-upper", "kind": "anchor", "label": "eq:fN-upper", "owner": "prop:dimension-improvement-structured-component", "section": "sec-structured-component", "anchor": { "source": "manuscript", "line_start": 3160, "line_end": 3161, "char_start": 190440, "char_end": 190459 }, "argument_gist": "", "excerpt": "_1 s'}, \\] then we are done as this establishes the goal above. Now suppose that |\\mathfrak{N}| < e^{d_i s' - 4 s' + m + 2 \\ell + 51 C_1 \\epsilon_1 s'}. On the other hand, by the definition of $\\fN$, we have that at least $\\beta$-proportional curve neighborhoods from $\\mathfrak{LL}$ are contained in some enlarged $(\\ell, m)$-neighborhood $L(\\tilde \\cC)$ where $\\t" }, { "id": "eq:decompose-hat-mu", "kind": "anchor", "label": "eq:decompose-hat-mu", "owner": "prop:high-dim-decomposition", "section": "sec-main-proof", "anchor": { "source": "manuscript", "line_start": 3249, "line_end": 3250, "char_start": 196674, "char_end": 196701 }, "argument_gist": "", "excerpt": "mu$ is $(4-\\theta, s')$-good. Then $a(2s')_\\ast \\hat \\mu$ can be written as a(2s')_\\ast \\hat \\mu = \\nu_{\\S} \\mu_{\\S} + \\sum_{k} \\nu_k a(2s')_\\ast \\mu_k, where $\\mu_{\\S}$ is a probability measure on $X$, $\\nu_{\\S} = O(\\beta)$, and for each $k$, $\\mu_k$ is a probability measure on $$\\exp\\left(B_{\\vv r^+}(\\beta)\\right)u\\left([\\beta]\\right)x_k$$" }, { "id": "eq:define-Upsilons'", "kind": "anchor", "label": "eq:define-Upsilons'", "owner": "prop:high-dim-decomposition", "section": "sec-main-proof", "anchor": { "source": "manuscript", "line_start": 3263, "line_end": 3264, "char_start": 197463, "char_end": 197490 }, "argument_gist": "", "excerpt": "ution-a} for the definition). \\par Let us define \\Upsilon_{s'} := B^{A_0 A U^\\ast}(\\beta) \\exp\\left(B_{\\vv r^-} (\\beta e^{-s'})\\right)\\exp\\left(B_{\\vv r^+} (\\beta)\\right) B^{U}(\\beta), and decompose $X_{\\beta^{1/4}}$ as \\[ X_{\\beta^{1/4}} = \\bigcup_{k=1}^{\\beta^{-100} e^{2s'}} \\Upsilon_{s'} x_k. \\] We say $x_k$ is \\" }, { "id": "eq:first-decompose-hat-mu", "kind": "anchor", "label": "eq:first-decompose-hat-mu", "owner": "prop:high-dim-decomposition", "section": "sec-main-proof", "anchor": { "source": "manuscript", "line_start": 3281, "line_end": 3282, "char_start": 198366, "char_end": 198399 }, "argument_gist": "", "excerpt": "( \\Upsilon_{\\mathrm{light}} \\right) \\le \\beta^{10}.\\] Therefore, we have \\hat \\mu = \\sum_{x_k \\text{ heavy }} \\bar{\\nu}_k \\bar{\\mu}_k +O(\\beta^{10}). We claim that for every heavy $x_k$, $\\bar \\mu_k$ is nice, namely, for $y \\in \\supp \\bar{\\mu}_k$, \\[ \\bar{\\mu}_k (Q_{s'}(\\beta) B^{A_0 H}(\\beta)y) \\le \\beta^2 e^{-2s' + 2\\theta s'}. \\] In" }, { "id": "eq:bary-y-Qs", "kind": "anchor", "label": "eq:bary-y-Qs", "owner": "prop:high-dim-decomposition", "section": "sec-main-proof", "anchor": { "source": "manuscript", "line_start": 3303, "line_end": 3304, "char_start": 199795, "char_end": 199815 }, "argument_gist": "", "excerpt": ":= \\exp(\\vv w^+) x_k.\\] Since $\\|\\vv w^-\\| \\le \\beta e^{-s'}$, we have that \\bar{y} \\in Q_{s'}(\\beta) B^{A_0 H}(\\beta)y. Moreover, we may apply \\eqref{eq:fvy}, \\eqref{eq:fwy}, and \\eqref{eq:fy} to conclude that for every $r' \\in [\\beta]$, there exists $f(r')$ such that $|r' - f(r')| \\le \\beta|r'|$, and \\[ a(2s')u\\left(f(r')\\right) \\bar{y} \\in B^G(\\beta) u" }, { "id": "eq:compare-u-orbit", "kind": "anchor", "label": "eq:compare-u-orbit", "owner": "prop:high-dim-decomposition", "section": "sec-main-proof", "anchor": { "source": "manuscript", "line_start": 3310, "line_end": 3310, "char_start": 200200, "char_end": 200226 }, "argument_gist": "", "excerpt": "in B^G(\\beta) u(r') y. \\] Therefore, for any function $f \\in C^{\\infty}_c(X)$, \\left| \\int_{[1]} f(a(2s')u(\\beta r) y) \\dd r - \\int_{[1]} f(a(2s')u(\\beta r) \\bar y) \\dd r\\right| \\ll \\beta \\|f\\|_{C^1}. Let us denote the union of $\\bar y$'s constructed as above by $\\mathcal T$ and define $\\mu_k$ as the normalized Lebesgue measure on the union $$ \\bigcup_{\\ba" }, { "id": "external:Bernik-Kleinbock-Margulis", "kind": "external", "label": "[BKM01] Bernik-Kleinbock-Margulis", "bibliography_key": "Bernik-Kleinbock-Margulis", "bibliography_number": "BKM01", "title": "V~Bernik, D~Kleinbock, and Gregory~A Margulis. \\newblock Khintchine-type theorems on manifolds: the convergence case for standard and multiplicative versions. \\newblock {\\em International Mathematics Research Notices}, 2001(9):453--486, 2001.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3358, "line_end": 3365, "char_start": 204246, "char_end": 204537 }, "review_reference_roles": [ { "consumer": "cor:curve", "role": "proof_input", "claimed_role": "proof_input", "audit_status": "not_inspected", "imported_result": "Theorem 1.4, invoked to obtain a uniform exponentially small bound for Leb(fm_ell). The exact external statement is not supplied.", "required_hypotheses": [ "The parameter-dependent lattice map must satisfy the theorem's quantitative goodness and nondegeneracy hypotheses.", "The required lower bounds on the relevant covolume functions must hold uniformly over the stated ell range.", "The theorem must control the compactness condition involving both vectors and exterior-square vectors." ] } ] }, { "id": "external:burger1990", "kind": "external", "label": "[Bur90] burger1990", "bibliography_key": "burger1990", "bibliography_number": "Bur90", "title": "Marc Burger. \\newblock Horocycle flow on geometrically finite surfaces. \\newblock {\\em Duke Mathematical Journal}, 61(3):779--803, 1990.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3365, "line_end": 3370, "char_start": 204537, "char_end": 204703 } }, { "id": "external:Browning2016", "kind": "external", "label": "[BV16] Browning2016", "bibliography_key": "Browning2016", "bibliography_number": "BV16", "title": "Tim Browning and Ilya Vinogradov. \\newblock Effective {Ratner} theorem for $\\operatorname{SL}(2,\\mathbb{R})\\ltimes\\mathbb{R}^{2}$ and gaps in $ \\sqrt{n}$ modulo $1$. \\newblock {\\em Journal of the London Mathematical Society}, 94(1):61--84, 2016.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3370, "line_end": 3377, "char_start": 204703, "char_end": 204983 } }, { "id": "external:chow-yang", "kind": "external", "label": "[CY24] chow-yang", "bibliography_key": "chow-yang", "bibliography_number": "CY24", "title": "Sam Chow and Lei Yang. \\newblock Effective equidistribution for multiplicative diophantine approximation on lines. \\newblock {\\em Inventiones mathematicae}, 235(3):973--1007, 2024.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3377, "line_end": 3383, "char_start": 204983, "char_end": 205193 } }, { "id": "external:Dani1981", "kind": "external", "label": "[Dan81] Dani1981", "bibliography_key": "Dani1981", "bibliography_number": "Dan81", "title": "S.~G. Dani. \\newblock Invariant measures and minimal sets of horospherical flows. \\newblock {\\em Inventiones mathematicae}, 64(2):357--385, 1981.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3383, "line_end": 3388, "char_start": 205193, "char_end": 205366 } }, { "id": "external:dani-smillie", "kind": "external", "label": "[DS84] dani-smillie", "bibliography_key": "dani-smillie", "bibliography_number": "DS84", "title": "Shrikrishna~G Dani and John Smillie. \\newblock Uniform distribution of horocycle orbits for fuchsian groups. \\newblock {\\em Duke Mathematical Journal}, 51(1):185--194, 1984.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3388, "line_end": 3393, "char_start": 205366, "char_end": 205570 } }, { "id": "external:edwards-effective", "kind": "external", "label": "[Edw21] edwards-effective", "bibliography_key": "edwards-effective", "bibliography_number": "Edw21", "title": "Samuel Edwards. \\newblock On the rate of equidistribution of expanding translates of horospheres in ${\\Gamma} \\setminus {G}$. \\newblock {\\em Commentarii Mathematici Helvetici}, 96(2):275--337, 2021.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3393, "line_end": 3399, "char_start": 205570, "char_end": 205807 } }, { "id": "external:einsiedler-mohammadi", "kind": "external", "label": "[EM22] einsiedler-mohammadi", "bibliography_key": "einsiedler-mohammadi", "bibliography_number": "EM22", "title": "M~Einsiedler and A~Mohammadi. \\newblock Effective arguments in unipotent dynamics. \\newblock {\\em Dynamics, geometry, number theory: the impact of margulis on modern mathematics}, 2022.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3399, "line_end": 3405, "char_start": 205807, "char_end": 206033 } }, { "id": "external:einsiedler-margulis-mohammadi-venkatesh", "kind": "external", "label": "[EMMV20] einsiedler-margulis-mohammadi-venkatesh", "bibliography_key": "einsiedler-margulis-mohammadi-venkatesh", "bibliography_number": "EMMV20", "title": "Einsiedler, Margulis, Mohammadi, and Venkatesh. \\newblock Effective equidistribution and property $(\\tau)$. \\newblock {\\em Journal of the AMS}, 33(1), 2020.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3405, "line_end": 3410, "char_start": 206033, "char_end": 206249 } }, { "id": "external:einsiedler-margulis-venkatesh", "kind": "external", "label": "[EMV09] einsiedler-margulis-venkatesh", "bibliography_key": "einsiedler-margulis-venkatesh", "bibliography_number": "EMV09", "title": "Manfred Einsiedler, Gregory Margulis, and Akshay Venkatesh. \\newblock Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces. \\newblock {\\em Inventiones mathematicae}, 177(1):137--212, 2009.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3410, "line_end": 3416, "char_start": 206249, "char_end": 206523 }, "review_reference_roles": [ { "consumer": "prop:initial-bound-dimension", "role": "proof_input", "claimed_role": "proof_input", "audit_status": "not_inspected", "imported_result": "An effective approximate-stabilizer-to-exact-stabilizer argument, used first for one integer matrix and then simultaneously for two noncommuting matrices. No theorem or lemma locator is supplied.", "required_hypotheses": [ "A specified representation and algebraic stabilizer.", "Height bounds for the integer matrices and the representative g.", "A quantitative distance-to-exact-stabilizer estimate with the stated exponents.", "For Case 2, a simultaneous estimate for both matrices." ] } ] }, { "id": "external:Flaminio-Forni", "kind": "external", "label": "[FF03] Flaminio-Forni", "bibliography_key": "Flaminio-Forni", "bibliography_number": "FF03", "title": "Livio Flaminio and Giovanni Forni. \\newblock Invariant distributions and time averages for horocycle flows. \\newblock {\\em Duke Math. J.}, 119(3):465--526, 2003.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3416, "line_end": 3421, "char_start": 206523, "char_end": 206717 } }, { "id": "external:furstenberg1973", "kind": "external", "label": "[Fur73] furstenberg1973", "bibliography_key": "furstenberg1973", "bibliography_number": "Fur73", "title": "Harry Furstenberg. \\newblock The unique ergodigity of the horocycle flow. \\newblock In {\\em Recent advances in topological dynamics}, pages 95--115. Springer, 1973.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3421, "line_end": 3427, "char_start": 206717, "char_end": 206918 } }, { "id": "external:green_tao2012", "kind": "external", "label": "[GT12] green_tao2012", "bibliography_key": "green_tao2012", "bibliography_number": "GT12", "title": "Ben Green and Terence Tao. \\newblock The quantitative behaviour of polynomial orbits on nilmanifolds. \\newblock {\\em Annals of Mathematics}, 175(2):465--540, 2012.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3427, "line_end": 3432, "char_start": 206918, "char_end": 207113 } }, { "id": "external:KNSY", "kind": "external", "label": "[KdSSY24] KNSY", "bibliography_key": "KNSY", "bibliography_number": "KdSSY24", "title": "Dmitry Kleinbock, Nicolas de~Saxc\\'e, Nimish~A. Shah, and Pengyu Yang. \\newblock Equidistribution in the space of 3-lattices and {D}irichlet-improvable vectors on planar lines. \\newblock {\\em Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)}, 25(1):565--604, 2024.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3432, "line_end": 3438, "char_start": 207113, "char_end": 207396 }, "review_reference_roles": [ { "consumer": "cor:straight-line", "role": "background", "claimed_role": "background", "audit_status": "not_inspected", "imported_result": "", "required_hypotheses": [] } ] }, { "id": "external:kim2021", "kind": "external", "label": "[Kim21] kim2021", "bibliography_key": "kim2021", "bibliography_number": "Kim21", "title": "Wooyeon Kim. \\newblock Effective equidistribution of expanding translates in the space of affine lattices. \\newblock {\\em arXiv preprint arXiv:2110.00706}, 2021.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3438, "line_end": 3444, "char_start": 207396, "char_end": 207586 } }, { "id": "external:kleinbock2008", "kind": "external", "label": "[Kle08] kleinbock2008", "bibliography_key": "kleinbock2008", "bibliography_number": "Kle08", "title": "Dmitry Kleinbock. \\newblock An extension of quantitative nondivergence and applications to diophantine exponents. \\newblock {\\em Transactions of the American Mathematical Society}, 360(12):6497--6523, 2008.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3444, "line_end": 3451, "char_start": 207586, "char_end": 207829 }, "review_reference_roles": [ { "consumer": "prop:quantitative-nondivergence", "role": "proof_input", "claimed_role": "proof_input", "audit_status": "not_inspected", "imported_result": "Theorem 2.2, invoked as quantitative non-divergence for the polynomial lattice map r -> a(s)u(r)g.", "required_hypotheses": [ "Uniform goodness of the relevant exterior-power covolume functions.", "A positive uniform lower bound for their suprema on the parameter interval.", "The theorem's lattice-rank and interval hypotheses." ] } ] }, { "id": "external:Klein_Mar", "kind": "external", "label": "[KM98] Klein_Mar", "bibliography_key": "Klein_Mar", "bibliography_number": "KM98", "title": "Dmitry Kleinbock and Gregory Margulis. \\newblock Flows on homogeneous spaces and diophantine approximation on manifolds. \\newblock {\\em Annals of Mathematics}, pages 339--360, 1998.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3451, "line_end": 3457, "char_start": 207829, "char_end": 208040 }, "review_reference_roles": [ { "consumer": "prop:quantitative-nondivergence", "role": "proof_input", "claimed_role": "proof_input", "audit_status": "not_inspected", "imported_result": "Theorem 5.2, cited as an alternative quantitative non-divergence input.", "required_hypotheses": [ "Goodness of all relevant primitive-subgroup covolume functions.", "Uniform lower bounds on their suprema.", "The applicable interval and lattice hypotheses." ] } ] }, { "id": "external:Klein-Mar-Effective-Equid", "kind": "external", "label": "[KM12] Klein-Mar-Effective-Equid", "bibliography_key": "Klein-Mar-Effective-Equid", "bibliography_number": "KM12", "title": "Dmitry Kleinbock and Gregory Margulis. \\newblock {\\em On effective equidistribution of expanding translates of certain orbits in the space of lattices}, pages 385--396. \\newblock Springer, Boston, MA, 2012.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3457, "line_end": 3463, "char_start": 208040, "char_end": 208292 } }, { "id": "external:lindenstrauss-margulis", "kind": "external", "label": "[LM14] lindenstrauss-margulis", "bibliography_key": "lindenstrauss-margulis", "bibliography_number": "LM14", "title": "Elon Lindenstrauss and Gregory Margulis. \\newblock Effective estimates on indefinite ternary forms. \\newblock {\\em Israel Journal of Mathematics}, 203(1):445--499, 2014.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3463, "line_end": 3468, "char_start": 208292, "char_end": 208502 } }, { "id": "external:LM-preprint2021", "kind": "external", "label": "[LM23] LM-preprint2021", "bibliography_key": "LM-preprint2021", "bibliography_number": "LM23", "title": "Elon Lindenstrauss and Amir Mohammadi. \\newblock Polynomial effective density in quotients of $\\mathbb{H}^3$ and $\\mathbb{H}^2 \\times \\mathbb{H}^2$. \\newblock {\\em Inventiones Mathematicae}, 231(3):1141--1237, 2023.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3468, "line_end": 3474, "char_start": 208502, "char_end": 208753 }, "review_reference_roles": [ { "consumer": "prop:high-dimension-to-equidistribution-a", "role": "proof_input", "claimed_role": "proof_input", "audit_status": "not_inspected", "imported_result": "Proposition 4.2 and its proof, claimed to transfer step by step to the present SL(3,R) action. The exact external theorem and the transfer are not supplied.", "required_hypotheses": [ "A suitable quantitative matrix-coefficient estimate for the present G action and Sobolev norm.", "The stated Frostman bound at scale beta e^{-s'}.", "Uniform control for base points in X_{beta^{1/4}}.", "Verification that the transverse directions, flow relations and parameter ranges in the cited proof apply here." ] } ] }, { "id": "external:LMW-preprint2022", "kind": "external", "label": "[LMW23] LMW-preprint2022", "bibliography_key": "LMW-preprint2022", "bibliography_number": "LMW23", "title": "Elon Lindenstrauss, Amir Mohammadi, and Zhiren Wang. \\newblock Polynomial effective equidistribution. \\newblock {\\em Comptes Rendus. Math{\\'e}matique}, 361(G2):507--520, 2023.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3474, "line_end": 3479, "char_start": 208753, "char_end": 208964 } }, { "id": "external:margulis-thesis", "kind": "external", "label": "[Mar04] margulis-thesis", "bibliography_key": "margulis-thesis", "bibliography_number": "Mar04", "title": "Gregory~Aleksandrovitsch Margulis. \\newblock On some aspects of the theory of anosov systems. \\newblock In {\\em On Some Aspects of the Theory of Anosov Systems}, pages 1--71. Springer, 2004.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3479, "line_end": 3485, "char_start": 208964, "char_end": 209191 } }, { "id": "external:Mozes_Shah", "kind": "external", "label": "[MS95] Mozes_Shah", "bibliography_key": "Mozes_Shah", "bibliography_number": "MS95", "title": "Shahar Mozes and Nimish Shah. \\newblock On the space of ergodic invariant measures of unipotent flows. \\newblock {\\em Ergodic Theory and Dynamical Systems}, 15(01):149--159, 1995.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3485, "line_end": 3490, "char_start": 209191, "char_end": 209399 } }, { "id": "external:nelson-venkatesh", "kind": "external", "label": "[NV21] nelson-venkatesh", "bibliography_key": "nelson-venkatesh", "bibliography_number": "NV21", "title": "P.~Nelson and A.~Venkatesh. \\newblock The orbit method and analysis of automorphic forms. \\newblock {\\em Acta Mathematica}, 226(1), 2021.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3490, "line_end": 3495, "char_start": 209399, "char_end": 209571 } }, { "id": "external:ratner-acta", "kind": "external", "label": "[Rat90a] ratner-acta", "bibliography_key": "ratner-acta", "bibliography_number": "Rat90a", "title": "Marina Ratner. \\newblock On measure rigidity of unipotent subgroups of semisimple groups. \\newblock {\\em Acta mathematica}, 165:229--309, 1990.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3495, "line_end": 3500, "char_start": 209571, "char_end": 209746 } }, { "id": "external:ratner-invent", "kind": "external", "label": "[Rat90b] ratner-invent", "bibliography_key": "ratner-invent", "bibliography_number": "Rat90b", "title": "Marina Ratner. \\newblock Strict measure rigidity for unipotent subgroups of solvable groups. \\newblock {\\em Inventiones mathematicae}, 101(1):449--482, 1990.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3500, "line_end": 3505, "char_start": 209746, "char_end": 209937 } }, { "id": "external:Ratner", "kind": "external", "label": "[Rat91a] Ratner", "bibliography_key": "Ratner", "bibliography_number": "Rat91a", "title": "Marina Ratner. \\newblock On {Raghunathan's} measure conjecture. \\newblock {\\em Annals of Mathematics}, pages 545--607, 1991.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3505, "line_end": 3510, "char_start": 209937, "char_end": 210088 } }, { "id": "external:ratner_2", "kind": "external", "label": "[Rat91b] ratner_2", "bibliography_key": "ratner_2", "bibliography_number": "Rat91b", "title": "Marina Ratner. \\newblock Raghunathan's topological conjecture and distributions of unipotent flows. \\newblock {\\em Duke Math. J}, 63(1):235--280, 1991.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3510, "line_end": 3516, "char_start": 210088, "char_end": 210270 }, "review_reference_roles": [ { "consumer": "thm:ratner", "role": "background", "claimed_role": "background", "audit_status": "not_inspected", "imported_result": "", "required_hypotheses": [] } ] }, { "id": "external:sarnak1982", "kind": "external", "label": "[Sar82] sarnak1982", "bibliography_key": "sarnak1982", "bibliography_number": "Sar82", "title": "Peter Sarnak. \\newblock Asymptotic behavior of periodic orbits of the horocycle flow and eisenstein series. \\newblock {\\em Communications on Pure and Applied Mathematics}, 34(6):719--739, 1982.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3516, "line_end": 3523, "char_start": 210270, "char_end": 210497 } }, { "id": "external:Shah_2", "kind": "external", "label": "[Sha09] Shah_2", "bibliography_key": "Shah_2", "bibliography_number": "Sha09", "title": "Nimish~A Shah. \\newblock Equidistribution of expanding translates of curves and {Dirichlet's} theorem on diophantine approximation. \\newblock {\\em Inventiones mathematicae}, 177(3):509--532, 2009.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3523, "line_end": 3529, "char_start": 210497, "char_end": 210721 }, "review_reference_roles": [ { "consumer": "cor:curve", "role": "background", "claimed_role": "background", "audit_status": "not_inspected", "imported_result": "", "required_hypotheses": [] } ] }, { "id": "external:Strombergsson2004", "kind": "external", "label": "[Str04] Strombergsson2004", "bibliography_key": "Strombergsson2004", "bibliography_number": "Str04", "title": "Andreas Str{\\\"o}mbergsson. \\newblock On the uniform equidistribution of long closed horocycles. \\newblock {\\em Duke Math. J.}, 123(3):507--547, 06 2004.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3529, "line_end": 3534, "char_start": 210721, "char_end": 210910 } }, { "id": "external:Strombergsson2013", "kind": "external", "label": "[Str13] Strombergsson2013", "bibliography_key": "Strombergsson2013", "bibliography_number": "Str13", "title": "Andreas Str{\\\"o}mbergsson. \\newblock On the deviation of ergodic averages for horocycle flows. \\newblock {\\em Journal of Modern Dynamics}, 7:291, 2013.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3534, "line_end": 3539, "char_start": 210910, "char_end": 211098 } }, { "id": "external:Strombergsson2015", "kind": "external", "label": "[Str15] Strombergsson2015", "bibliography_key": "Strombergsson2015", "bibliography_number": "Str15", "title": "Andreas Str{\\\"o}mbergsson. \\newblock An effective {Ratner} equidistribution result for $\\operatorname{SL}(2,\\mathbb{R})\\ltimes\\mathbb{R}^{2}$. \\newblock {\\em Duke Mathematical Journal}, 164(5):843--902, 04 2015.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3539, "line_end": 3545, "char_start": 211098, "char_end": 211348 } }, { "id": "external:Sarnak-Ubis", "kind": "external", "label": "[SU15] Sarnak-Ubis", "bibliography_key": "Sarnak-Ubis", "bibliography_number": "SU15", "title": "Peter Sarnak and Adri{\\'a}n Ubis. \\newblock The horocycle flow at prime times. \\newblock {\\em Journal de Math\u00e9matiques Pures et Appliqu\u00e9es}, 103(2):575--618, 2015.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3545, "line_end": 3551, "char_start": 211348, "char_end": 211543 } }, { "id": "external:Venkatesh2010", "kind": "external", "label": "[Ven10] Venkatesh2010", "bibliography_key": "Venkatesh2010", "bibliography_number": "Ven10", "title": "A.~Venkatesh. \\newblock Sparse equidistribution problems, period bounds and subconvexity. \\newblock {\\em Annals of Mathematics}, 172:989--1094, 2010.", "urls": [], "argument_gist": "", "anchor": { "source": "manuscript", "line_start": 3551, "line_end": 3556, "char_start": 211543, "char_end": 211725 }, "review_reference_roles": [ { "consumer": "prop:high-dimension-to-equidistribution-a", "role": "proof_input", "claimed_role": "proof_input", "audit_status": "not_inspected", "imported_result": "The van der Corput-type equidistribution argument in Section 3, cited as the method underlying this proposition. No exact transferable theorem is identified.", "required_hypotheses": [ "Quantitative mixing and the regularity bounds needed by the averaging argument.", "Control of correlations for the particular transverse measure and unipotent subgroup used here." ] } ] } ], "edges": [ { "from": "thm:effective-U-orbit", "to": "sec-intro", "type": "internal_reference", "evidence": "not\\in X_{e^{-|\\ell|/3}}$ for some $ 0.99 t \\le |\\ell| \\le 1.01 t$. \\par Proving Theorem \\ref{thm:effective-U-orbit} is equivalent to proving the following theorem: Let $\\eta, t_0 >0$ be as above. For any $t", "anchor": { "source": "manuscript", "line_start": 489, "line_end": 489, "char_start": 17824, "char_end": 17851 }, "origin": "source_reference", "context": "narrative" }, { "from": "thm:main-thm", "to": "sec-intro", "type": "internal_reference", "evidence": "u(r) x_0 ) \\dd r = \\int_{[1]} f(u(r e^t )y_0) \\dd r, \\] where $y_0 = a(t)x_0$. \\par Using Theorem \\ref{thm:main-thm}, we can prove the following two corollaries on effective equidistribution of expanding curves under translates of diagonal subgr", "anchor": { "source": "manuscript", "line_start": 501, "line_end": 501, "char_start": 18487, "char_end": 18505 }, "origin": "source_reference", "context": "narrative" }, { "from": "cor:straight-line", "to": "rmk:expanding-line", "type": "internal_reference", "evidence": "of. The qualitative version of Corollary \\ref{cor:straight-line} was proved in \\cite{KNSY} for $\\omega(\\vv w) \\le 2$ using Ratner's theorem. The authors also proved that the equidistribution", "anchor": { "source": "manuscript", "line_start": 550, "line_end": 550, "char_start": 20849, "char_end": 20872 }, "origin": "source_reference", "context": "statement" }, { "from": "cor:straight-line", "to": "cor:curve", "type": "internal_reference", "evidence": "Let us use the same notation as in Corollary \\ref{cor:straight-line}. Let $\\psi : [1] \\to \\R^2$ be a smooth non-degenerate curve in $\\R^2$, namely, derivatives of $\\psi$ span the whole space $\\R^", "anchor": { "source": "manuscript", "line_start": 560, "line_end": 560, "char_start": 21340, "char_end": 21363 }, "origin": "source_reference", "context": "statement" }, { "from": "cor:straight-line", "to": "thm:main-thm", "type": "candidate_proof_dependency", "evidence": "egin{bmatrix} 1 & \\psi_2'(r_0) & \\\\ & 1 & \\\\ & & 1 ,\\] as defined in Corollary \\ref{cor:straight-line}, and \\[ x_0 = a_0(t/2) a_1(t/2) z(-\\psi_2'(r_0)) n(\\psi(r_0)) \\Gamma. \\] Then if we can show that \\[ \\left\\{ a(t/2) u(r) x", "anchor": { "source": "manuscript", "line_start": 576, "line_end": 576, "char_start": 22312, "char_end": 22335 }, "origin": "source_reference", "context": "proof" }, { "from": "cor:curve", "to": "rmk:expanding-curve", "type": "internal_reference", "evidence": "f. The ineffective version of Corollary \\ref{cor:curve} was proved in \\cite{Shah_2} using Ratner's theorem. \\subsection*{Acknowledgements} The author thanks Wen Huang,", "anchor": { "source": "manuscript", "line_start": 601, "line_end": 601, "char_start": 23803, "char_end": 23818 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:define-a", "to": "sec-prelim", "type": "internal_reference", "evidence": ": t \\in \\R \\right\\} \\subset Z_G(H). \\] Note that the definitions of $a(t)$ and $a_0(t)$ here match \\eqref{eq:define-a} and \\eqref{eq:define-a0}, respectively. The adjoint action of $a_0(t)$ on $\\fg$ is as follows: \\[ \\Ad (a_0(t)) \\bh = \\bh, \\text{", "anchor": { "source": "manuscript", "line_start": 698, "line_end": 698, "char_start": 27781, "char_end": 27800 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:define-a0", "to": "sec-prelim", "type": "internal_reference", "evidence": "set Z_G(H). \\] Note that the definitions of $a(t)$ and $a_0(t)$ here match \\eqref{eq:define-a} and \\eqref{eq:define-a0}, respectively. The adjoint action of $a_0(t)$ on $\\fg$ is as follows: \\[ \\Ad (a_0(t)) \\bh = \\bh, \\text{ for } \\bh \\in \\fh, \\] \\[\\", "anchor": { "source": "manuscript", "line_start": 699, "line_end": 699, "char_start": 27806, "char_end": 27826 }, "origin": "source_reference", "context": "narrative" }, { "from": "lm:lie-bracket", "to": "lm:product-lie-algebra-1", "type": "candidate_proof_dependency", "evidence": "+ \\left[ \\vv h, \\left[\\vv h, \\vv w\\right] \\right]\\right) + \\cdots. By Lemma \\ref{lm:lie-bracket} and the fact that for any $\\vv w_1, \\vv w_2 \\in \\fg$, $\\|[\\vv w_1, \\vv w_2]\\| \\le \\|\\vv w_1\\| \\cdot \\|\\vv w_2\\|$, we have t", "anchor": { "source": "manuscript", "line_start": 779, "line_end": 779, "char_start": 31614, "char_end": 31634 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:product-w-h-1", "to": "lm:product-lie-algebra-1", "type": "candidate_proof_dependency", "evidence": "\\|$, and $\\bar{\\vv h} \\in \\R \\fa_0 + \\fh$ with $\\|\\bar{\\vv h}\\| \\le 2\\rho_1 \\|\\vv h\\|$. This proves \\eqref{eq:product-w-h-1}. \\par Conversely, for $\\tilde{\\vv w} \\in B_{\\fr_1 +\\fr_2}(\\rho_1)$ and $\\tilde{\\vv h} \\in B_{\\R \\fa_0 + \\fh}(\\rho_2)$, let u", "anchor": { "source": "manuscript", "line_start": 784, "line_end": 784, "char_start": 32318, "char_end": 32342 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:product-w-h-1", "to": "lm:product-lie-algebra-1", "type": "candidate_proof_dependency", "evidence": "where $\\vv w \\in B_{\\fr_1 + \\fr_2}(10^{-10})$ and $\\vv h \\in B_{\\R \\fa_0 + \\fh}(10^{-10})$. Then by \\eqref{eq:product-w-h-1}, we have \\[ \\| \\vv w - \\tilde{\\vv w} \\| \\le 2 \\|\\vv h\\| \\|\\vv w\\|, \\text{ and } \\| \\vv h - \\tilde{\\vv h} \\| \\le 2 \\|\\vv w\\", "anchor": { "source": "manuscript", "line_start": 787, "line_end": 787, "char_start": 32660, "char_end": 32684 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:product-w-h-2", "to": "lm:product-lie-algebra-1", "type": "candidate_proof_dependency", "evidence": "w}\\|, \\text{ and } \\| \\vv h - \\tilde{\\vv h} \\| \\le 4 \\rho_1 \\|\\tilde{\\vv h}\\|. \\] This proves \\eqref{eq:product-w-h-2}. Let $0 < \\rho < \\rho_0 < 10^{-10}$ be two small constants. Let $\\vv w, \\vv w'", "anchor": { "source": "manuscript", "line_start": 795, "line_end": 795, "char_start": 33184, "char_end": 33208 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:lie-bracket", "to": "lm:product-w", "type": "candidate_proof_dependency", "evidence": "t[ \\vv w', \\left[\\vv w', \\vv w\\right] \\right]\\right) + \\cdots. \\par By Lemma \\ref{lm:lie-bracket}, we have the following observations: \\par For $i = 2j+1$, every monomial $[\\cdots\\diamond\\cdots]$ in $P(\\vv w, \\vv w')$ with", "anchor": { "source": "manuscript", "line_start": 812, "line_end": 812, "char_start": 34106, "char_end": 34126 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:product-lie-algebra-1", "to": "lm:product-w", "type": "candidate_proof_dependency", "evidence": "It is easy to see that \\[ \\|\\hat{\\vv w}\\| \\le 3 \\rho_0. \\] By Lemma \\ref{lm:product-lie-algebra-1}, we have that \\[ \\exp(\\hat{\\vv w} + \\bar{\\vv h}) = \\exp(\\tilde{\\vv w}) \\exp(\\vv h) \\] where $\\tilde{\\vv w} \\in \\fr_1 + \\", "anchor": { "source": "manuscript", "line_start": 833, "line_end": 833, "char_start": 35564, "char_end": 35594 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:hatw-ww'", "to": "lm:product-w", "type": "candidate_proof_dependency", "evidence": "rho, and $\\vv h \\in \\R \\fa_0 + \\fh$ with $\\|\\vv h\\| \\le 4 \\rho$. Noting that \\eqref{eq:hatw-ww'} and \\eqref{eq:tildew-hatw} imply that \\[ \\| \\tilde{\\vv w} - \\vv w - \\vv w'\\| \\le 26 \\rho_0 \\rho, \\] we conclude the lemm", "anchor": { "source": "manuscript", "line_start": 840, "line_end": 840, "char_start": 35969, "char_end": 35988 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:tildew-hatw", "to": "lm:product-w", "type": "candidate_proof_dependency", "evidence": "and $\\vv h \\in \\R \\fa_0 + \\fh$ with $\\|\\vv h\\| \\le 4 \\rho$. Noting that \\eqref{eq:hatw-ww'} and \\eqref{eq:tildew-hatw} imply that \\[ \\| \\tilde{\\vv w} - \\vv w - \\vv w'\\| \\le 26 \\rho_0 \\rho, \\] we conclude the lemma. \\subse", "anchor": { "source": "manuscript", "line_start": 840, "line_end": 840, "char_start": 35993, "char_end": 36015 }, "origin": "source_reference", "context": "proof" }, { "from": "sec-initial-dim-control", "to": "sec-prelim", "type": "internal_reference", "evidence": "owever, compared with previous works, we take a quite different approach in this paper. \\par In \\S \\ref{sec-initial-dim-control}, we We will start with $\\cF_0 = a(s)u([1])x_0$ for $s = t - \\delta_1 t$ and study the dimension of the normalized Lebesgue meas", "anchor": { "source": "manuscript", "line_start": 852, "line_end": 852, "char_start": 36570, "char_end": 36599 }, "origin": "source_reference", "context": "narrative" }, { "from": "thm:main-thm", "to": "sec-prelim", "type": "internal_reference", "evidence": "\\cF_0$ has certain dimension control in the transversal direction unless condition (2) in Theorem \\ref{thm:main-thm} holds (see Proposition \\ref{prop:initial-bound-dimension}). Here we are allowed to remove an exponentially small proportion fro", "anchor": { "source": "manuscript", "line_start": 854, "line_end": 854, "char_start": 36935, "char_end": 36953 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:initial-bound-dimension", "to": "sec-prelim", "type": "internal_reference", "evidence": "the transversal direction unless condition (2) in Theorem \\ref{thm:main-thm} holds (see Proposition \\ref{prop:initial-bound-dimension}). Here we are allowed to remove an exponentially small proportion from $\\cF_0$. At this step quantitative non-divergence resu", "anchor": { "source": "manuscript", "line_start": 854, "line_end": 854, "char_start": 36977, "char_end": 37011 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec-quantitative-nondivergence", "to": "sec-prelim", "type": "internal_reference", "evidence": "ally small proportion from $\\cF_0$. At this step quantitative non-divergence results proved in \\S \\ref{sec-quantitative-nondivergence} are needed. This is the starting point of our proof. The argument in this part is similar to the corresponding parts in \\cite{", "anchor": { "source": "manuscript", "line_start": 856, "line_end": 856, "char_start": 37158, "char_end": 37194 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec-reduction", "to": "sec-prelim", "type": "internal_reference", "evidence": "art is similar to the corresponding parts in \\cite{LM-preprint2021,LMW-preprint2022}. \\par \\S \\ref{sec-reduction} to \\S \\ref{sec-structured-component} is the crucial part. In these sections, we will construct a sequence $\\{\\cF_i : i \\in \\N\\}$", "anchor": { "source": "manuscript", "line_start": 860, "line_end": 860, "char_start": 37371, "char_end": 37390 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec-structured-component", "to": "sec-prelim", "type": "internal_reference", "evidence": "rresponding parts in \\cite{LM-preprint2021,LMW-preprint2022}. \\par \\S \\ref{sec-reduction} to \\S \\ref{sec-structured-component} is the crucial part. In these sections, we will construct a sequence $\\{\\cF_i : i \\in \\N\\}$ of $U$-orbits starting from $\\cF_0$", "anchor": { "source": "manuscript", "line_start": 860, "line_end": 860, "char_start": 37397, "char_end": 37427 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:improve-dimension", "to": "sec-prelim", "type": "internal_reference", "evidence": "arting from $\\cF_0$ where $\\cF_{i+1} = a(s_i)\\cF_i$ for each $i \\ge 0$. We will prove Proposition \\ref{prop:improve-dimension} which says that $\\mu_{\\cF_{i+1}}$ has a better dimension control compared with $\\mu_{\\cF_i}$ at a larger scale as a cost. To", "anchor": { "source": "manuscript", "line_start": 862, "line_end": 862, "char_start": 37637, "char_end": 37665 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec-kakeya-model", "to": "sec-prelim", "type": "internal_reference", "evidence": "r dimension control compared with $\\mu_{\\cF_i}$ at a larger scale as a cost. To prove this, in \\S \\ref{sec-kakeya-model} we introduce a Kakeya-type model to study the behavior of $U$-orbits in a neighborhood. The proof of Proposition \\ref{prop:impr", "anchor": { "source": "manuscript", "line_start": 864, "line_end": 864, "char_start": 37813, "char_end": 37835 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:improve-dimension", "to": "sec-prelim", "type": "internal_reference", "evidence": "a Kakeya-type model to study the behavior of $U$-orbits in a neighborhood. The proof of Proposition \\ref{prop:improve-dimension} is divided into three parts: Proposition \\ref{prop:dimension-improvement-random}, Proposition \\ref{prop:dimension-improvement-str", "anchor": { "source": "manuscript", "line_start": 865, "line_end": 865, "char_start": 37951, "char_end": 37979 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:dimension-improvement-random", "to": "sec-prelim", "type": "internal_reference", "evidence": "ood. The proof of Proposition \\ref{prop:improve-dimension} is divided into three parts: Proposition \\ref{prop:dimension-improvement-random}, Proposition \\ref{prop:dimension-improvement-strange-part} from \\S \\ref{sec-dimension-improvement-unstructured}, and Proposition", "anchor": { "source": "manuscript", "line_start": 865, "line_end": 865, "char_start": 38021, "char_end": 38060 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:dimension-improvement-strange-part", "to": "sec-prelim", "type": "internal_reference", "evidence": "sion} is divided into three parts: Proposition \\ref{prop:dimension-improvement-random}, Proposition \\ref{prop:dimension-improvement-strange-part} from \\S \\ref{sec-dimension-improvement-unstructured}, and Proposition \\ref{prop:dimension-improvement-structured-component} from", "anchor": { "source": "manuscript", "line_start": 865, "line_end": 865, "char_start": 38074, "char_end": 38119 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec-dimension-improvement-unstructured", "to": "sec-prelim", "type": "internal_reference", "evidence": "op:dimension-improvement-random}, Proposition \\ref{prop:dimension-improvement-strange-part} from \\S \\ref{sec-dimension-improvement-unstructured}, and Proposition \\ref{prop:dimension-improvement-structured-component} from \\S \\ref{sec-structured-component}. The key of the pr", "anchor": { "source": "manuscript", "line_start": 865, "line_end": 865, "char_start": 38128, "char_end": 38172 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:dimension-improvement-structured-component", "to": "sec-prelim", "type": "internal_reference", "evidence": "ion-improvement-strange-part} from \\S \\ref{sec-dimension-improvement-unstructured}, and Proposition \\ref{prop:dimension-improvement-structured-component} from \\S \\ref{sec-structured-component}. The key of the proofs is to calculate the weighted intersection number of the Kakeya-typ", "anchor": { "source": "manuscript", "line_start": 865, "line_end": 865, "char_start": 38190, "char_end": 38243 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec-structured-component", "to": "sec-prelim", "type": "internal_reference", "evidence": "vement-unstructured}, and Proposition \\ref{prop:dimension-improvement-structured-component} from \\S \\ref{sec-structured-component}. The key of the proofs is to calculate the weighted intersection number of the Kakeya-type model. The final outcome of the calcu", "anchor": { "source": "manuscript", "line_start": 865, "line_end": 865, "char_start": 38252, "char_end": 38282 }, "origin": "source_reference", "context": "narrative" }, { "from": "lm:curve-in-N", "to": "sec-prelim", "type": "internal_reference", "evidence": "ghborhood of a piece of an orbit of a subgroup isomorphic to $\\SL(2, \\R) \\ltimes \\R^2$ (see Lemma \\ref{lm:curve-in-N}). The latter case can be ruled out by our Diophantine condition (see Proposition \\ref{prop:sl2xr2-closing-lemma}). At each st", "anchor": { "source": "manuscript", "line_start": 868, "line_end": 868, "char_start": 38639, "char_end": 38658 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:sl2xr2-closing-lemma", "to": "sec-prelim", "type": "internal_reference", "evidence": "f{lm:curve-in-N}). The latter case can be ruled out by our Diophantine condition (see Proposition \\ref{prop:sl2xr2-closing-lemma}). At each step we are allowed to remove an exponentially small proportion. This part is novel compared with previous works. \\p", "anchor": { "source": "manuscript", "line_start": 869, "line_end": 869, "char_start": 38742, "char_end": 38773 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:high-dimension-to-equidistribution-a", "to": "sec-prelim", "type": "internal_reference", "evidence": "tions transversal to the $A_0 H$-orbit direction is almost optimal. Then we can apply Proposition \\ref{prop:high-dimension-to-equidistribution-a} from \\S \\ref{sec-high-dim-to-equidistribution} to conclude the effective equidistribution. Proposition \\ref{prop:high-dimension", "anchor": { "source": "manuscript", "line_start": 873, "line_end": 873, "char_start": 39140, "char_end": 39187 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec-high-dim-to-equidistribution", "to": "sec-prelim", "type": "internal_reference", "evidence": "st optimal. Then we can apply Proposition \\ref{prop:high-dimension-to-equidistribution-a} from \\S \\ref{sec-high-dim-to-equidistribution} to conclude the effective equidistribution. Proposition \\ref{prop:high-dimension-to-equidistribution-a} can be proved by foll", "anchor": { "source": "manuscript", "line_start": 873, "line_end": 873, "char_start": 39196, "char_end": 39234 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:high-dimension-to-equidistribution-a", "to": "sec-prelim", "type": "internal_reference", "evidence": "\\S \\ref{sec-high-dim-to-equidistribution} to conclude the effective equidistribution. Proposition \\ref{prop:high-dimension-to-equidistribution-a} can be proved by following a Van der Corput type argument due to Venkatesh.", "anchor": { "source": "manuscript", "line_start": 874, "line_end": 874, "char_start": 39293, "char_end": 39340 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:initial-bound-dimension", "to": "sec-quantitative-nondivergence", "type": "internal_reference", "evidence": "non-divergence result and a quantitative separation lemma, which will be used to prove Proposition \\ref{prop:initial-bound-dimension}. There exists $ \\alpha > 0$ such that the following holds: For any $\\", "anchor": { "source": "manuscript", "line_start": 917, "line_end": 917, "char_start": 42124, "char_end": 42158 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:high-dimension-to-equidistribution-a", "to": "subsec-constants", "type": "internal_reference", "evidence": ", $d_0 = 10^{-6}$, $C_1 = 10 d_0^{-1} = 10^7$. \\item Let $\\theta >0$ be the constant from Theorem \\ref{prop:high-dimension-to-equidistribution-a}. Without loss of generality, we can assume that $\\theta < 10^{-10}$. \\item Let $\\alpha>0$ be the constants as in Proposition \\r", "anchor": { "source": "manuscript", "line_start": 980, "line_end": 980, "char_start": 46134, "char_end": 46181 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:quantitative-nondivergence", "to": "subsec-constants", "type": "internal_reference", "evidence": ", we can assume that $\\theta < 10^{-10}$. \\item Let $\\alpha>0$ be the constants as in Proposition \\ref{prop:quantitative-nondivergence} and $\\alpha_2 = \\alpha/4$. \\item $\\epsilon_2 = 10^{-20} d_0 \\theta$, $N_1 = 10^5 \\epsilon_2^{-1}$, $N_2 = 10^6$, $\\epsilon = e", "anchor": { "source": "manuscript", "line_start": 981, "line_end": 981, "char_start": 46309, "char_end": 46346 }, "origin": "source_reference", "context": "narrative" }, { "from": "thm:main-thm", "to": "sec-initial-dim-control", "type": "internal_reference", "evidence": "n} and $Q_s(\\bar \\beta) := Q_s^s(\\bar \\beta)$. \\par Let $x_{0} \\in X$ be such that (2) in Theorem \\ref{thm:main-thm} does not hold. Let us start with the normalized measure on $\\cE = a(s) u([1])x_{0}$ where $s = (1 -\\delta_1) t >0$. This is a $", "anchor": { "source": "manuscript", "line_start": 1029, "line_end": 1029, "char_start": 48847, "char_end": 48865 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:quantitative-nondivergence", "to": "sec-initial-dim-control", "type": "internal_reference", "evidence": "$. Let $\\mu_{\\cF}$ denote the normalized $U$-orbit measure on $\\cF$. \\par By Proposition \\ref{prop:quantitative-nondivergence} and our assumption on $x_{\\mathrm{initial}}$, it is easy to show that the removed proportion is $O(s^2 \\beta^{ \\alpha_2})$. Th", "anchor": { "source": "manuscript", "line_start": 1039, "line_end": 1039, "char_start": 49458, "char_end": 49495 }, "origin": "source_reference", "context": "narrative" }, { "from": "lm:separation-lemma-a", "to": "prop:initial-bound-dimension", "type": "candidate_proof_dependency", "evidence": "satisfies $\\bar x \\in X_{\\beta^{1/4}}$ and $a(-2 d_0 \\ell) \\bar x \\in X_{\\beta^{1/4}}$, by Lemma \\ref{lm:separation-lemma-a}, if we write $h_{\\bar x} = \\exp(\\vv h_{\\bar x})$, and \\[ \\vv h_{\\bar x} = a_0 \\fa + a_1 \\fu + a_2 \\fu^\\ast,\\] we will have", "anchor": { "source": "manuscript", "line_start": 1070, "line_end": 1070, "char_start": 51175, "char_end": 51202 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:a0ge", "to": "prop:initial-bound-dimension", "type": "candidate_proof_dependency", "evidence": "whose $\\fu$ coordinate is $a_1 + a_0 L + a_2 L^2$. Since $|a_2| \\ge \\beta e^{-2 d_0 \\ell}$ (by \\eqref{eq:a0ge}) and $|L| \\ge e^{5 d_0 \\ell}$, we have that \\[ |a_1 + a_0 L + a_2 L^2| \\ge \\beta e^{8 d_0 \\ell}. \\] Therefore, we h", "anchor": { "source": "manuscript", "line_start": 1110, "line_end": 1110, "char_start": 54107, "char_end": 54122 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:bound-3", "to": "prop:initial-bound-dimension", "type": "candidate_proof_dependency", "evidence": "\\[ |a_1 + a_0 L + a_2 L^2| \\ge \\beta e^{8 d_0 \\ell}. \\] Therefore, we have that the equality \\eqref{eq:bound-3} is impossible to hold and conclude the claim. \\par Let us consider the adjoint action of $G$ on $\\bigoplus_{j=1}^8 \\wedge^j", "anchor": { "source": "manuscript", "line_start": 1113, "line_end": 1113, "char_start": 54274, "char_end": 54292 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:bound-2", "to": "prop:initial-bound-dimension", "type": "candidate_proof_dependency", "evidence": "\\fa \\wedge \\fu \\wedge \\fu^{\\ast}.$$ Then the stabilizer of $v_H$ \\[\\Stab(v_H) = A_0 H.\\] Then \\eqref{eq:bound-2} implies that \\[ \\gamma_{\\bar x} g^{-1} v_H = \\exp(\\Ad ( g^{-1})\\vv v'_{\\bar x}) g^{-1} v_H. \\] The norm of $g^{-1} v_H$ is", "anchor": { "source": "manuscript", "line_start": 1119, "line_end": 1119, "char_start": 54549, "char_end": 54567 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:initial-bound-dimension", "to": "prop:sl2xr2-closing-lemma", "type": "candidate_proof_dependency", "evidence": "ell = \\ell/2 \\in [\\delta_3 t, \\delta_1 t]$. By repeating the argument in the proof of Proposition \\ref{prop:initial-bound-dimension}, we can show that $a_0(\\eta_3 \\tilde \\ell) \\tilde x = a_{0}(\\ell +\\eta_3 \\tilde \\ell)x \\not\\in X_{\\beta^{1/4}}$, which contra", "anchor": { "source": "manuscript", "line_start": 1185, "line_end": 1185, "char_start": 59916, "char_end": 59950 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:initial-bound-dimension", "to": "sec-reduction", "type": "internal_reference", "evidence": "ood for any $\\ell \\in [\\ell_1 , \\ell_2]$. \\par According to this definition, Proposition \\ref{prop:initial-bound-dimension} says that $\\mu_{\\cF}$ is $( d_0, [\\delta_3 t, \\delta_1 t])$-good. \\par Starting from $s_0 = \\delta_1 t/2$ and $d_0$, let us def", "anchor": { "source": "manuscript", "line_start": 1209, "line_end": 1209, "char_start": 61597, "char_end": 61631 }, "origin": "source_reference", "context": "narrative" }, { "from": "def:initial-construction", "to": "sec-reduction", "type": "internal_reference", "evidence": "ove that $\\mu_{\\cF_{i+1}}$ has better dimension control than $\\mu_{\\cF_i}$. Similarly to Definition \\ref{def:initial-construction}, by removing an exponentially small proportion, we can assume that for any $x \\in \\supp \\mu_{\\cF_i}$, any $0 \\le \\ell \\le 2 \\delt", "anchor": { "source": "manuscript", "line_start": 1216, "line_end": 1216, "char_start": 62184, "char_end": 62214 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:initial-bound-dimension", "to": "sec-reduction", "type": "internal_reference", "evidence": "le 2 \\delta_1 t$, $a_0(\\ell')a(-\\ell)x \\in X_{\\beta^{1/4}}$. By repeating the proof of Proposition \\ref{prop:initial-bound-dimension} and \\ref{prop:sl2xr2-closing-lemma}, we have that Proposition \\ref{prop:initial-bound-dimension} and \\ref{prop:sl2xr2-closing-lem", "anchor": { "source": "manuscript", "line_start": 1219, "line_end": 1219, "char_start": 62483, "char_end": 62517 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:sl2xr2-closing-lemma", "to": "sec-reduction", "type": "internal_reference", "evidence": "\\in X_{\\beta^{1/4}}$. By repeating the proof of Proposition \\ref{prop:initial-bound-dimension} and \\ref{prop:sl2xr2-closing-lemma}, we have that Proposition \\ref{prop:initial-bound-dimension} and \\ref{prop:sl2xr2-closing-lemma} hold for $\\mu_{\\cF_i}$. \\par For", "anchor": { "source": "manuscript", "line_start": 1219, "line_end": 1219, "char_start": 62522, "char_end": 62553 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:initial-bound-dimension", "to": "sec-reduction", "type": "internal_reference", "evidence": "n \\ref{prop:initial-bound-dimension} and \\ref{prop:sl2xr2-closing-lemma}, we have that Proposition \\ref{prop:initial-bound-dimension} and \\ref{prop:sl2xr2-closing-lemma} hold for $\\mu_{\\cF_i}$. \\par For notational simplicity, for $s' >0, \\ell \\ge 0$, let us deno", "anchor": { "source": "manuscript", "line_start": 1219, "line_end": 1219, "char_start": 62580, "char_end": 62614 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:sl2xr2-closing-lemma", "to": "sec-reduction", "type": "internal_reference", "evidence": "nd \\ref{prop:sl2xr2-closing-lemma}, we have that Proposition \\ref{prop:initial-bound-dimension} and \\ref{prop:sl2xr2-closing-lemma} hold for $\\mu_{\\cF_i}$. \\par For notational simplicity, for $s' >0, \\ell \\ge 0$, let us denote \\label{eq:pi-", "anchor": { "source": "manuscript", "line_start": 1219, "line_end": 1219, "char_start": 62619, "char_end": 62650 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:reduction", "to": "sec-reduction", "type": "internal_reference", "evidence": "his completes the proof. \\par Let us denote $\\epsilon_1 = 2 \\epsilon_2$. By Proposition \\ref{prop:reduction}, it suffices to prove that for any full playground $\\Pi(x)$, the measure $\\mu_{\\Pi(x)}$ is $(d_i+ \\epsilon_1, s_{i+1})$-good aft", "anchor": { "source": "manuscript", "line_start": 1276, "line_end": 1276, "char_start": 65775, "char_end": 65795 }, "origin": "source_reference", "context": "narrative" }, { "from": "lm:product-w", "to": "lm:two-theta", "type": "candidate_proof_dependency", "evidence": "et us take $$g = a_0 a u^\\ast (r^\\ast) \\exp(\\vv w^-) \\exp(\\vv w^+) u(r) \\in \\Theta_{s'}.$$ By Lemma \\ref{lm:product-w}, \\[\\exp(\\vv w^-) \\exp(\\vv w^+) = \\exp(\\bar{\\vv w} ) \\exp( \\bar{\\vv h}), \\] where $\\bar{\\vv w} \\in B_{\\fr_1 + \\fr_2}(1.01\\beta e^", "anchor": { "source": "manuscript", "line_start": 1324, "line_end": 1324, "char_start": 68291, "char_end": 68309 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:two-theta", "to": "lm:product-theta", "type": "candidate_proof_dependency", "evidence": "^{-1} \\Theta_{s'} \\subset \\bar{\\Theta}_{s'}. \\] Other cases can be proved similarly. \\par By Lemma \\ref{lm:two-theta}, it suffices to show that \\[ \\hat{\\Theta}_{s'}^{-1} \\hat{\\Theta}_{s'} \\subset \\bar{\\Theta}_{s'},\\] where $\\hat{\\Theta}_{s'}$ is", "anchor": { "source": "manuscript", "line_start": 1342, "line_end": 1342, "char_start": 69133, "char_end": 69151 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:two-theta", "to": "lm:product-theta", "type": "candidate_proof_dependency", "evidence": "1} \\hat{\\Theta}_{s'} \\subset \\bar{\\Theta}_{s'},\\] where $\\hat{\\Theta}_{s'}$ is defined as in Lemma \\ref{lm:two-theta}. \\par Let us take $g = \\exp(\\vv w) \\exp(\\vv h) \\in \\hat{\\Theta}_{s'}$ and $g' = \\exp(\\vv w') \\exp(\\vv h') \\in \\hat{\\Theta}_{s'}$.", "anchor": { "source": "manuscript", "line_start": 1344, "line_end": 1344, "char_start": 69302, "char_end": 69320 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:product-w", "to": "lm:product-theta", "type": "candidate_proof_dependency", "evidence": "\\Theta}_{s'}$. Then \\[ g^{-1} g' = \\exp(-\\vv h) \\exp(-\\vv w) \\exp(\\vv w') \\exp(\\vv h'). \\] By Lemma \\ref{lm:product-w}, we have \\[ \\exp(-\\vv w) \\exp(\\vv w') = \\exp(\\bar{\\vv w}) \\exp (\\bar{\\vv h}), \\] where $\\bar{\\vv w} \\in B_{\\fr_1 + \\fr_2}(2.201\\", "anchor": { "source": "manuscript", "line_start": 1347, "line_end": 1347, "char_start": 69536, "char_end": 69554 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:product-theta", "to": "sec-kakeya-model", "type": "internal_reference", "evidence": "conclude that $g^{-1} g' \\in \\bar{\\Theta}_{s'}$. This completes the proof. Using Lemma \\ref{lm:product-theta}, it is easy to prove the following: \\par Let us denote \\[ \\tilde{\\Theta}_{s'}", "anchor": { "source": "manuscript", "line_start": 1354, "line_end": 1354, "char_start": 69998, "char_end": 70020 }, "origin": "source_reference", "context": "narrative" }, { "from": "lm:product-theta", "to": "cor:product-theta", "type": "candidate_proof_dependency", "evidence": "\\Theta_{s'} \\subset \\tilde{\\Theta}_{s'}. \\] Other cases can be proved similarly. In fact, by Lemma \\ref{lm:product-theta}, we have \\[ \\Theta_{s'}\\Theta_{s'} \\subset \\bar{\\Theta}_{s'}, \\] where $\\bar{\\Theta}_{s'}$ is defined as in Lemma \\ref{lm:produc", "anchor": { "source": "manuscript", "line_start": 1365, "line_end": 1365, "char_start": 70583, "char_end": 70605 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:product-theta", "to": "cor:product-theta", "type": "candidate_proof_dependency", "evidence": "heta_{s'}\\Theta_{s'} \\subset \\bar{\\Theta}_{s'}, \\] where $\\bar{\\Theta}_{s'}$ is defined as in Lemma \\ref{lm:product-theta}. By repeating the proof of Lemma \\ref{lm:product-theta}, we get \\[ \\bar{\\Theta}_{s'} \\bar{\\Theta}_{s'} \\subset \\tilde{\\Theta}_{s", "anchor": { "source": "manuscript", "line_start": 1367, "line_end": 1367, "char_start": 70721, "char_end": 70743 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:product-theta", "to": "cor:product-theta", "type": "candidate_proof_dependency", "evidence": "$\\bar{\\Theta}_{s'}$ is defined as in Lemma \\ref{lm:product-theta}. By repeating the proof of Lemma \\ref{lm:product-theta}, we get \\[ \\bar{\\Theta}_{s'} \\bar{\\Theta}_{s'} \\subset \\tilde{\\Theta}_{s'}. \\] Combining the two inclusions above, we conclude t", "anchor": { "source": "manuscript", "line_start": 1367, "line_end": 1367, "char_start": 70777, "char_end": 70799 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:product-w", "to": "lm:different-coordinate-1", "type": "candidate_proof_dependency", "evidence": "r{\\vv w}^+]\\| \\le \\|\\bar{\\vv w}^-\\| \\cdot \\|\\bar{\\vv w}^+\\| \\le 0.001 \\beta e^{-2\\ell}, \\] by Lemma \\ref{lm:product-w}, we have \\[ \\exp(\\bar{\\vv w}^-) \\exp(\\bar{\\vv w}^+) = \\exp(\\tilde{\\vv w}^- + \\tilde{\\vv w}^+) \\exp( \\vv h), \\] where $\\tilde{\\vv", "anchor": { "source": "manuscript", "line_start": 1392, "line_end": 1392, "char_start": 72678, "char_end": 72696 }, "origin": "source_reference", "context": "proof" }, { "from": "def:kakeya-model", "to": "sec-kakeya-model", "type": "internal_reference", "evidence": "ta \\le |J_1| \\le |J_2| \\le 4 \\beta$. \\par We will postpone the proof after Definition \\ref{def:kakeya-model}. \\par For $\\bar x \\in \\Sigma^r \\subset \\Pi(x)$, let us denote \\[ \\Pi_{s', \\ell} (\\bar x) = \\Pi_{s', \\ell} \\bar x, \\] and \\[\\S", "anchor": { "source": "manuscript", "line_start": 1421, "line_end": 1421, "char_start": 74665, "char_end": 74687 }, "origin": "source_reference", "context": "narrative" }, { "from": "lm:different-coordinate-1", "to": "lm:initial-dimension-under-a-t", "type": "candidate_proof_dependency", "evidence": "ft( B_{\\vv r^+}(\\beta e^{-\\ell}) \\right) B^{U}(\\beta) x' \\] where $x' = a(-2s') \\bar x$. By Lemma \\ref{lm:different-coordinate-1}, we have that \\[ B^{A_0 A}(\\beta) B^{U^\\ast}(\\beta) \\exp\\left( B_{\\vv r^-}(\\beta e^{-\\ell}) \\right) \\exp\\left( B_{\\vv r^+}(\\be", "anchor": { "source": "manuscript", "line_start": 1442, "line_end": 1442, "char_start": 75957, "char_end": 75988 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:initial-bound-dimension", "to": "lm:initial-dimension-under-a-t", "type": "candidate_proof_dependency", "evidence": "B^{U}(\\beta) \\subset Q_\\ell(2\\beta) B^{A_0 H }(2\\beta).\\] By repeating the proof of Proposition \\ref{prop:initial-bound-dimension} to $\\cF_i$ we have \\[ \\mu_{\\cF_i}(Q_\\ell(2\\beta) B^{A_0 H }(2\\beta) x') \\le e^{-d_0 \\ell}, \\] which concludes the proof. \\en", "anchor": { "source": "manuscript", "line_start": 1444, "line_end": 1444, "char_start": 76239, "char_end": 76273 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:fy", "to": "def:kakeya-model", "type": "internal_reference", "evidence": "frac{ r (r w_{2,y} v_{2,y})^2 }{4 - (r w_{2,y} v_{2,y})^2} \\par From \\eqref{eq:fy} it is easy to see that $|f_{y} (r) - r_y - r| \\le \\beta |r|$. \\par Let $\\cT(y)$ denote the $2\\beta e^{-s'}$-neighborhood of $\\cL", "anchor": { "source": "manuscript", "line_start": 1522, "line_end": 1522, "char_start": 80913, "char_end": 80926 }, "origin": "source_reference", "context": "statement" }, { "from": "lm:cross-section", "to": "sec-kakeya-model", "type": "internal_reference", "evidence": "\\cW(y)\\}$ the Kakeya-type model for $(\\tilde \\mu, \\Pi(x))$. \\par Now let us prove Lemma \\ref{lm:cross-section}: [Proof of Lemma \\ref{lm:cross-section}] \\par Without loss of generality, we can assume $r=0$. We first prove $J_0", "anchor": { "source": "manuscript", "line_start": 1539, "line_end": 1539, "char_start": 81989, "char_end": 82011 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:fy", "to": "lm:cross-section", "type": "candidate_proof_dependency", "evidence": "= \\tilde r(r)$ is a smooth function of $r$ satisfies that $|r-\\tilde r| \\le \\beta|r|$. Moreover, by \\eqref{eq:fy}, we have \\[ u(\\tilde r) \\exp(\\vv w_y^-) \\exp(\\vv w_y^+) u(r_y) x = a_{0,y,\\tilde r} a_{y,\\tilde r} u^\\ast (r^\\ast_{y,\\tilde r}", "anchor": { "source": "manuscript", "line_start": 1549, "line_end": 1549, "char_start": 82988, "char_end": 83001 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:product-theta", "to": "lm:partition-1", "type": "candidate_proof_dependency", "evidence": "we have \\[ z' \\in \\Theta^{-1}_{s'} \\tilde z \\subset \\Theta^{-1}_{s'} \\Theta_{s'} z. \\] By Lemma \\ref{lm:product-theta}, we have \\[ z' \\in \\bar{\\Theta}_{s'}z, \\] namely, \\[ z' = \\exp(\\bar{\\vv w}) \\exp(\\bar{\\vv h}) z, \\] where $\\bar{\\vv w} \\in B_{\\", "anchor": { "source": "manuscript", "line_start": 1642, "line_end": 1642, "char_start": 89462, "char_end": 89484 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:different-coordinate-1", "to": "lm:partition-1", "type": "candidate_proof_dependency", "evidence": "^\\ast \\in [2.22\\beta e^{-2s'}]$, and $\\vv w'' \\in B_{\\fr_1 + \\fr_2} (2.22\\beta e^{-s'})$. By Lemma \\ref{lm:different-coordinate-1}, we have \\[ \\exp(\\vv w'' ) = \\exp(\\tilde b \\fa + \\tilde b_0 \\fa_0 + \\tilde r^\\ast \\fu^\\ast) \\exp(\\vv w^-) \\exp(\\vv w^+) u(\\tilde", "anchor": { "source": "manuscript", "line_start": 1649, "line_end": 1649, "char_start": 89955, "char_end": 89986 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:z-z'", "to": "lm:partition-1", "type": "candidate_proof_dependency", "evidence": "})$, and $\\vv w^+ \\in B_{\\vv r^+}(2.3 \\beta e^{-s'})$. Let us consider $\\bP(z)$ and $\\bP(z')$. From \\eqref{eq:z-z'}, we have that \\[ \\bP(z) \\in \\bP(z') + \\left[3.1\\beta\\right] \\times \\left[2.3 \\beta e^{-s'}\\right]^2. \\] From the definition of $\\", "anchor": { "source": "manuscript", "line_start": 1655, "line_end": 1655, "char_start": 90672, "char_end": 90687 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:partition-2", "to": "sec-kakeya-model", "type": "internal_reference", "evidence": "z$ such that \\[ \\Omega(y) \\cap \\Omega(y') \\neq \\emptyset. \\] \\par Before proving Lemma \\ref{lm:partition-2}, let us introduce a coordinate system in $\\Theta(z)$: \\par Given $z \\in \\cI$ and $ y \\", "anchor": { "source": "manuscript", "line_start": 1667, "line_end": 1667, "char_start": 91224, "char_end": 91244 }, "origin": "source_reference", "context": "narrative" }, { "from": "lm:partition-2", "to": "sec-kakeya-model", "type": "internal_reference", "evidence": "w_2 \\in \\vv r^-$. Then we define \\[ \\fD_z(y) := (v_2, w_2). \\] \\par Let us prove Lemma \\ref{lm:partition-2}: [Proof of Lemma \\ref{lm:partition-2}] \\par We first prove that for any $z' \\in \\Theta'(z)$, there exists $y \\in \\cI", "anchor": { "source": "manuscript", "line_start": 1675, "line_end": 1675, "char_start": 91754, "char_end": 91774 }, "origin": "source_reference", "context": "narrative" }, { "from": "lm:two-theta", "to": "lm:partition-2", "type": "candidate_proof_dependency", "evidence": "that for any $z' \\in \\Theta'(z)$, there exists $y \\in \\cI_z$ such that $z' \\in \\Omega(y)$. By Lemma \\ref{lm:two-theta}, $\\Theta_{s'} \\subset \\hat{\\Theta}_{s'}$. Therefore, we can write \\[ z' = a_0 a u(r) \\exp(\\vv w^+ + \\vv w^-)u^\\ast (r^\\ast) z,", "anchor": { "source": "manuscript", "line_start": 1677, "line_end": 1677, "char_start": 91947, "char_end": 91965 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:product-w", "to": "lm:partition-2", "type": "candidate_proof_dependency", "evidence": "= a_0 a u(r) \\exp(\\vv w^+ + \\vv w^-) \\exp(-\\bar{\\vv w}^-) u^\\ast (r^\\ast)y. By Lemma \\ref{lm:product-w}, we have that \\[ \\exp(\\vv w^+ + \\vv w^-) \\exp(-\\bar{\\vv w}^-) = \\exp(\\vv w^+ + \\vv w^- - \\bar{\\vv w}^- + \\tilde{\\vv w} ) \\exp(\\", "anchor": { "source": "manuscript", "line_start": 1688, "line_end": 1688, "char_start": 92640, "char_end": 92658 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:z'-y", "to": "lm:partition-2", "type": "candidate_proof_dependency", "evidence": "ve estimates imply that $\\|p_{-} (\\vv w)\\| \\le \\beta e^{-3s'}$. By plugging the above equation into \\eqref{eq:z'-y}, we get that $z' \\in \\Omega(y)$. \\par Let us prove that there are at most $N_2$ different $y \\in \\cI_z$ such that \\[ \\Omega(z')", "anchor": { "source": "manuscript", "line_start": 1695, "line_end": 1695, "char_start": 93245, "char_end": 93260 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:product-w", "to": "lm:partition-2", "type": "candidate_proof_dependency", "evidence": "vv h') \\exp( - \\tilde{\\vv w}^+ - \\tilde{\\vv w}^-) \\exp(\\vv w^+ + \\vv w^-) \\exp(\\vv h) y.\\] By Lemma \\ref{lm:product-w}, \\[ \\exp( - \\tilde{\\vv w}^+ - \\tilde{\\vv w}^-) \\exp(\\vv w^+ + \\vv w^-) = \\exp(\\tilde{\\vv w}^+ + \\tilde{\\vv w}^- ) \\exp( \\tilde{\\", "anchor": { "source": "manuscript", "line_start": 1708, "line_end": 1708, "char_start": 93932, "char_end": 93950 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:different-coordinate-1", "to": "lm:partition-2", "type": "candidate_proof_dependency", "evidence": "v^+ \\in B_{\\vv r^+} (4.3\\beta e^{-s'})$, and $\\vv v^-\\in B_{\\vv r^-} (4.3\\beta e^{-3s'})$. By Lemma \\ref{lm:different-coordinate-1}, \\[ \\exp( \\vv v^+ + \\vv v^- ) = \\tilde{a}_0 \\tilde{a} u(\\tilde r) \\exp(\\tilde{\\vv v}^+)\\exp(\\tilde{\\vv v}^-) u^\\ast(\\tilde{r}^\\", "anchor": { "source": "manuscript", "line_start": 1716, "line_end": 1716, "char_start": 94685, "char_end": 94716 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:partition-2", "to": "rmk:partition-2", "type": "internal_reference", "evidence": "uation} \\fC := \\left[1.2 \\beta e^{-s'}\\right]^2. From Lemma \\ref{lm:partition-2}, we can see that for any $z' \\in \\Theta(z)$, \\[ \\fD_z(z') \\in \\fC. \\] \\par Su", "anchor": { "source": "manuscript", "line_start": 1733, "line_end": 1733, "char_start": 95713, "char_end": 95733 }, "origin": "source_reference", "context": "statement" }, { "from": "def:kakeya-model", "to": "lm:omega-u", "type": "internal_reference", "evidence": "ary time change: Let $\\tilde \\mu$ and $\\Pi(x)$ be as in Definition \\ref{def:kakeya-model}. Let us denote \\hat{\\Omega}_{s'} := Q^{s'}_{3s'}(2.1\\beta) Q^{A_0 H} (2s', 2.1\\bet", "anchor": { "source": "manuscript", "line_start": 1765, "line_end": 1765, "char_start": 97059, "char_end": 97081 }, "origin": "source_reference", "context": "statement" }, { "from": "lm:cross-section", "to": "lm:omega-u", "type": "candidate_proof_dependency", "evidence": "Also note that $u(r') y'$ and $u(r) y$ are in the same cross-section $\\Sigma^{\\tilde{r}}$. By Lemma \\ref{lm:cross-section}, $u(\\fl(y'))y' \\cap \\Sigma^{\\tilde{r}}$ is contained in an interval of length $\\asymp \\beta$. This implies that for any $y' \\in \\", "anchor": { "source": "manuscript", "line_start": 1794, "line_end": 1794, "char_start": 98754, "char_end": 98776 }, "origin": "source_reference", "context": "proof" }, { "from": "def:kakeya-model", "to": "lm:time-leave", "type": "internal_reference", "evidence": "d{proof} \\par Let $s' >0$ and $\\Pi(x) $ be as in Definition \\ref{def:kakeya-model}. Let us fix a cross-section $\\Sigma^r \\subset \\Pi(x)$ defined as above. Then for any $y_1, y_2 \\in \\fJ \\cap \\Sigma^r$, suppose th", "anchor": { "source": "manuscript", "line_start": 1806, "line_end": 1806, "char_start": 99472, "char_end": 99494 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:omega-y-r", "to": "lm:time-leave", "type": "internal_reference", "evidence": "\\cap \\Theta'(z) \\neq \\emptyset, \\text{ for } i=1,2, \\] where $\\Omega(y_i, L_i)$ is defined as in \\eqref{eq:omega-y-r}. \\par Let us denote $\\tilde{y}_i = u(L_i) y_i$ and write \\[ \\tilde{y}_1 = a_0 a u^\\ast(r^\\ast) \\exp(\\vv w^-) \\exp(\\vv w^+)", "anchor": { "source": "manuscript", "line_start": 1808, "line_end": 1808, "char_start": 99793, "char_end": 99813 }, "origin": "source_reference", "context": "statement" }, { "from": "lm:omega-u", "to": "lm:time-leave", "type": "candidate_proof_dependency", "evidence": "\\{ \\beta e^{2s'}, \\beta e^{-s'} \\|\\vv w^-\\|^{-1}\\right\\}.$$ \\par By Lemma \\ref{lm:omega-u}, we have $\\Omega(y_i, L_i) \\subset \\hat{\\Omega}(\\tilde{y}_i)$. Therefore, we have $ \\hat{\\Omega}(\\tilde{y}_1) \\cap \\Theta(z) \\neq", "anchor": { "source": "manuscript", "line_start": 1823, "line_end": 1823, "char_start": 100585, "char_end": 100601 }, "origin": "source_reference", "context": "proof" }, { "from": "cor:product-theta", "to": "lm:time-leave", "type": "candidate_proof_dependency", "evidence": "ilde{y}_1 \\in \\Theta_{s'}^{-1} \\Theta_{s'}\\Theta_{s'}^{-1} \\Theta_{s'} \\tilde{y}_2. \\] By Corollary \\ref{cor:product-theta}, \\[ \\tilde{y}_1 \\in \\tilde{\\Theta}_{s'} \\tilde{y}_2.\\] Then we have \\[ \\tilde{y}_1 = \\tilde{a}_0 \\tilde{a} u^{\\ast}(\\tilde{r}^\\a", "anchor": { "source": "manuscript", "line_start": 1829, "line_end": 1829, "char_start": 101176, "char_end": 101199 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:different-coordinate-1", "to": "lm:time-leave", "type": "candidate_proof_dependency", "evidence": "v w}^- ) u(\\tilde{r}) \\tilde{y}_2, \\] where $\\vv w^+ \\in B_{\\vv r^+}(4.5 \\beta e^{-s'})$. By Lemma \\ref{lm:different-coordinate-1}, if we write \\[ \\tilde{y}_1 = a_0 a u^\\ast(r^\\ast) \\exp(\\vv w^-) \\exp(\\vv w^+) u(r_0) \\tilde{y}_2, \\] we will have $\\|\\vv w^+ -", "anchor": { "source": "manuscript", "line_start": 1833, "line_end": 1833, "char_start": 101465, "char_end": 101496 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:fvy", "to": "lm:time-leave", "type": "candidate_proof_dependency", "evidence": "0 A}(\\beta)$, $u_1^{\\ast} \\in B^{U^\\ast}(\\beta e^{-2s'})$, and $|r_1 - r'_1| \\le \\beta |r_1|$. By \\eqref{eq:fvy}, \\eqref{eq:fwy}, and \\eqref{eq:fy}, we have that u(r_1) \\tilde{y}_1 &= \\bar a \\bar a_0 \\bar u^{\\ast } \\ex", "anchor": { "source": "manuscript", "line_start": 1843, "line_end": 1843, "char_start": 102158, "char_end": 102172 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:fwy", "to": "lm:time-leave", "type": "candidate_proof_dependency", "evidence": "_1^{\\ast} \\in B^{U^\\ast}(\\beta e^{-2s'})$, and $|r_1 - r'_1| \\le \\beta |r_1|$. By \\eqref{eq:fvy}, \\eqref{eq:fwy}, and \\eqref{eq:fy}, we have that u(r_1) \\tilde{y}_1 &= \\bar a \\bar a_0 \\bar u^{\\ast } \\exp(\\vv w^-(r'_1))", "anchor": { "source": "manuscript", "line_start": 1843, "line_end": 1843, "char_start": 102174, "char_end": 102188 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:fy", "to": "lm:time-leave", "type": "candidate_proof_dependency", "evidence": "ast}(\\beta e^{-2s'})$, and $|r_1 - r'_1| \\le \\beta |r_1|$. By \\eqref{eq:fvy}, \\eqref{eq:fwy}, and \\eqref{eq:fy}, we have that u(r_1) \\tilde{y}_1 &= \\bar a \\bar a_0 \\bar u^{\\ast } \\exp(\\vv w^-(r'_1)) \\exp(\\vv w^+(r'_1)", "anchor": { "source": "manuscript", "line_start": 1843, "line_end": 1843, "char_start": 102194, "char_end": 102207 }, "origin": "source_reference", "context": "proof" }, { "from": "def:coordinate-theta", "to": "sec-kakeya-model", "type": "internal_reference", "evidence": "\\beta e^{-s'}|w_2|^{-1}$. \\par This completes the proof. \\par Recall that in Definition \\ref{def:coordinate-theta}, we define a coordinate system in $\\Theta'(z)$ for any $z \\in \\cI$. By Remark \\ref{rmk:partition-2}, for any $z' \\in \\Theta'(z)$,", "anchor": { "source": "manuscript", "line_start": 1859, "line_end": 1859, "char_start": 103161, "char_end": 103187 }, "origin": "source_reference", "context": "narrative" }, { "from": "rmk:partition-2", "to": "sec-kakeya-model", "type": "internal_reference", "evidence": "def:coordinate-theta}, we define a coordinate system in $\\Theta'(z)$ for any $z \\in \\cI$. By Remark \\ref{rmk:partition-2}, for any $z' \\in \\Theta'(z)$, its coordinate $\\fD_z(z')$ is contained in \\[ \\fC = \\left[1.2 \\beta e^{-s'}\\right]^2.\\] The follow", "anchor": { "source": "manuscript", "line_start": 1859, "line_end": 1859, "char_start": 103266, "char_end": 103287 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:Iy1y2", "to": "lm:concentrated-surface", "type": "internal_reference", "evidence": ": \\par Recall that $I(y_1, y_2)$ is defined as in \\eqref{eq:Iy1y2}. Let us fix a cross-section $\\Sigma^r$. Let $z \\in \\cI\\cap \\Sigma^r, y \\in \\fJ\\cap \\Sigma^r$. Then the set of all coordinates $\\f", "anchor": { "source": "manuscript", "line_start": 1864, "line_end": 1864, "char_start": 103592, "char_end": 103608 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:fvy", "to": "lm:concentrated-surface", "type": "candidate_proof_dependency", "evidence": "where $\\vv w^- = v_2 \\vv v_2 + w_2 \\vv w_2$) satisfies that $I(\\bar z, y) \\neq \\emptyset$. Then by \\eqref{eq:fvy}, \\eqref{eq:fwy}, and \\eqref{eq:fy}, we have that \\[ u(r) \\bar z = a_{0,r} a_r u^{\\ast} (f^\\ast(r)) \\exp(\\vv f_-(r)) \\exp(\\vv f_+", "anchor": { "source": "manuscript", "line_start": 1890, "line_end": 1890, "char_start": 105417, "char_end": 105431 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:fwy", "to": "lm:concentrated-surface", "type": "candidate_proof_dependency", "evidence": "v_2 \\vv v_2 + w_2 \\vv w_2$) satisfies that $I(\\bar z, y) \\neq \\emptyset$. Then by \\eqref{eq:fvy}, \\eqref{eq:fwy}, and \\eqref{eq:fy}, we have that \\[ u(r) \\bar z = a_{0,r} a_r u^{\\ast} (f^\\ast(r)) \\exp(\\vv f_-(r)) \\exp(\\vv f_+(r)) u(f(r))y',", "anchor": { "source": "manuscript", "line_start": 1890, "line_end": 1890, "char_start": 105433, "char_end": 105447 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:fy", "to": "lm:concentrated-surface", "type": "candidate_proof_dependency", "evidence": "vv w_2$) satisfies that $I(\\bar z, y) \\neq \\emptyset$. Then by \\eqref{eq:fvy}, \\eqref{eq:fwy}, and \\eqref{eq:fy}, we have that \\[ u(r) \\bar z = a_{0,r} a_r u^{\\ast} (f^\\ast(r)) \\exp(\\vv f_-(r)) \\exp(\\vv f_+(r)) u(f(r))y', \\] where $\\vv f_+(", "anchor": { "source": "manuscript", "line_start": 1890, "line_end": 1890, "char_start": 105453, "char_end": 105466 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:time-leave", "to": "lm:concentrated-surface", "type": "candidate_proof_dependency", "evidence": "r v_{2} w_{2}} + v_{1},\\] and \\[ f_w(r) = \\frac{-2r w_{2}}{2 - r v_{2} w_{2}} + w_{1}.\\] By Lemma \\ref{lm:time-leave}, $ r \\in I(\\bar z, y)$ implies that $|f_v(r)|, |f_w(r)| \\le 4.6 \\beta e^{-s'}$. Let us denote $\\Delta = v_2 w_2/2$. Then we have", "anchor": { "source": "manuscript", "line_start": 1896, "line_end": 1896, "char_start": 105769, "char_end": 105788 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:1", "to": "lm:concentrated-surface", "type": "candidate_proof_dependency", "evidence": "This implies that $$\\|(v_2, w_2)\\| \\le \\beta^{-100}e^{-2s'} \\|(v_1, w_1)\\| .$$ Moreover, \\eqref{eq:1} and \\eqref{eq:2} imply that \\[ v_2 w_2 + v_1^{-1} v_2 + w_1^{-1} w_2 = O(\\beta^{-99} e^{-3s'} v_1^{-1}) + O(\\beta^{-99} e^{-3s'}", "anchor": { "source": "manuscript", "line_start": 1917, "line_end": 1917, "char_start": 106617, "char_end": 106629 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:2", "to": "lm:concentrated-surface", "type": "candidate_proof_dependency", "evidence": "at $$\\|(v_2, w_2)\\| \\le \\beta^{-100}e^{-2s'} \\|(v_1, w_1)\\| .$$ Moreover, \\eqref{eq:1} and \\eqref{eq:2} imply that \\[ v_2 w_2 + v_1^{-1} v_2 + w_1^{-1} w_2 = O(\\beta^{-99} e^{-3s'} v_1^{-1}) + O(\\beta^{-99} e^{-3s'} w_1^{-1}). \\]", "anchor": { "source": "manuscript", "line_start": 1917, "line_end": 1917, "char_start": 106634, "char_end": 106646 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:2", "to": "lm:concentrated-surface", "type": "candidate_proof_dependency", "evidence": "w^+) u(r_{y'}) y'\\] where $\\vv w^- = v_2 \\vv v_2 + w_2 \\vv w_2 \\in B_{\\vv r^-}(e^{-s'-j})$. Then by \\eqref{eq:2}, \\eqref{eq:1}, and \\eqref{eq:3}, we have that \\[ |v_2/v_1|, |w_2/ w_1| \\ge \\beta^{-1}e^{-2s'}. \\] This implies that \\[ \\|(v_1,", "anchor": { "source": "manuscript", "line_start": 1924, "line_end": 1924, "char_start": 107310, "char_end": 107322 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:1", "to": "lm:concentrated-surface", "type": "candidate_proof_dependency", "evidence": "y'\\] where $\\vv w^- = v_2 \\vv v_2 + w_2 \\vv w_2 \\in B_{\\vv r^-}(e^{-s'-j})$. Then by \\eqref{eq:2}, \\eqref{eq:1}, and \\eqref{eq:3}, we have that \\[ |v_2/v_1|, |w_2/ w_1| \\ge \\beta^{-1}e^{-2s'}. \\] This implies that \\[ \\|(v_1, w_1)\\| \\le \\be", "anchor": { "source": "manuscript", "line_start": 1924, "line_end": 1924, "char_start": 107324, "char_end": 107336 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:3", "to": "lm:concentrated-surface", "type": "candidate_proof_dependency", "evidence": "^- = v_2 \\vv v_2 + w_2 \\vv w_2 \\in B_{\\vv r^-}(e^{-s'-j})$. Then by \\eqref{eq:2}, \\eqref{eq:1}, and \\eqref{eq:3}, we have that \\[ |v_2/v_1|, |w_2/ w_1| \\ge \\beta^{-1}e^{-2s'}. \\] This implies that \\[ \\|(v_1, w_1)\\| \\le \\beta e^{2s'} \\|(v_2,", "anchor": { "source": "manuscript", "line_start": 1924, "line_end": 1924, "char_start": 107342, "char_end": 107354 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:hyperbola-box", "to": "lm:concentrated-surface", "type": "candidate_proof_dependency", "evidence": "This implies that \\[ \\|(v_1, w_1)\\| \\le \\beta e^{2s'} \\|(v_2, w_2)\\| \\le \\beta^2 e^{s' -j}. \\] By \\eqref{eq:hyperbola-box}, we have that the size of $\\fB(\\cT(y), z)$ is $\\le \\beta^{-98} e^{s'-j}$. This completes the proof.", "anchor": { "source": "manuscript", "line_start": 1928, "line_end": 1928, "char_start": 107521, "char_end": 107545 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:hyperbola", "to": "rmk:concentrated-surface", "type": "internal_reference", "evidence": "m:line} If $|v_1/w_1| \\le e^{-s'}$ or $|w_1/v_1| \\le e^{-s'}$, then the neighborhood described in~\\eqref{eq:hyperbola} agrees with the $ \\beta^{-100} e^{-3s'}$ neighborhood of the straight line \\[ v_1^{-1}(X - X_0) + w_1^{-1}(Y - Y_0) = 0.\\]", "anchor": { "source": "manuscript", "line_start": 1941, "line_end": 1941, "char_start": 108390, "char_end": 108410 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:hyperbola", "to": "sec-kakeya-model", "type": "internal_reference", "evidence": "e} \\medskip \\par Let us denote the $\\beta^{-100} e^{-3s'}$-neighborhood of the curve \\eqref{eq:hyperbola} by \\[\\cC(\\cT(y), z) \\subset \\fC.\\] By misuse of notation, we will write $\\Omega (\\bar z) \\subset \\cC(\\cT(y), z)$ when we actually", "anchor": { "source": "manuscript", "line_start": 1948, "line_end": 1948, "char_start": 108653, "char_end": 108673 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec-no-loss-of-dimension", "to": "sec-dimension-improvement-unstructured", "type": "internal_reference", "evidence": "\\noindent The proof of this proposition occupies the rest of this section as well as \\S \\ref{sec-no-loss-of-dimension} and \\S \\ref{sec-structured-component}. \\medskip \\par Let us define the heavy part of $\\Pi(x)$: \\label{def:heavy-pa", "anchor": { "source": "manuscript", "line_start": 1973, "line_end": 1973, "char_start": 110356, "char_end": 110386 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec-structured-component", "to": "sec-dimension-improvement-unstructured", "type": "internal_reference", "evidence": "s proposition occupies the rest of this section as well as \\S \\ref{sec-no-loss-of-dimension} and \\S \\ref{sec-structured-component}. \\medskip \\par Let us define the heavy part of $\\Pi(x)$: \\par For $z \\in \\cI$, we call", "anchor": { "source": "manuscript", "line_start": 1973, "line_end": 1973, "char_start": 110394, "char_end": 110424 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:improve-dimension", "to": "sec-dimension-improvement-unstructured", "type": "internal_reference", "evidence": "$\\cR_i$ will be said to be the heavy part of $\\Pi(x)$. \\par To prove Proposition \\ref{prop:improve-dimension}, it suffices to show that \\tilde \\mu(\\cR_i) \\le \\beta^{20}. \\end{equatio", "anchor": { "source": "manuscript", "line_start": 1988, "line_end": 1988, "char_start": 110987, "char_end": 111015 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:heavy-part-is-small", "to": "sec-dimension-improvement-unstructured", "type": "internal_reference", "evidence": "lon_1, s')$-good, which is what we wanted. \\par For the rest of the proof, we will focus on proving \\eqref{eq:heavy-part-is-small}. \\par We first prove the following lemma: Let us denote $s' = s_{i+1}$. For any $", "anchor": { "source": "manuscript", "line_start": 1996, "line_end": 1996, "char_start": 111817, "char_end": 111847 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:upper-omega", "to": "lm:omega-upper", "type": "candidate_proof_dependency", "evidence": "hich implies the claim. \\par Since $\\Pi(x)$ is a full playground, we have that for any $y \\in \\fJ$, \\eqref{eq:upper-omega} holds. \\par Let us further decompose $\\cR_i$ as follows: \\p", "anchor": { "source": "manuscript", "line_start": 2017, "line_end": 2017, "char_start": 112786, "char_end": 112808 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:upper-omega", "to": "def:heavy-part-decomposition", "type": "internal_reference", "evidence": "et \\Xi_j(z')$. Here $j \\le 2s'$ because for $j > 2s'$, \\[ \\Xi_j (z') \\subset \\Omega(z'), \\] and by \\eqref{eq:upper-omega}, \\[ \\tilde \\mu (\\Omega(z')) \\le \\beta^{-15}e^{-2 s'} e^{-2 d_i s'}, \\] which implies (since $\\Theta'(z)$ is heavy) that \\eqref{e", "anchor": { "source": "manuscript", "line_start": 2043, "line_end": 2043, "char_start": 114258, "char_end": 114280 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:strange-nbhd-condition", "to": "def:heavy-part-decomposition", "type": "internal_reference", "evidence": "ga(z')) \\le \\beta^{-15}e^{-2 s'} e^{-2 d_i s'}, \\] which implies (since $\\Theta'(z)$ is heavy) that \\eqref{eq:strange-nbhd-condition} can not hold for $\\Xi_j(z')$ and~$\\Theta'(z)$.", "anchor": { "source": "manuscript", "line_start": 2045, "line_end": 2045, "char_start": 114402, "char_end": 114435 }, "origin": "source_reference", "context": "statement" }, { "from": "def:kakeya-model", "to": "def:structure-randomness", "type": "internal_reference", "evidence": "\\par Let $s'>0$, $\\Pi(x)$ and $\\tilde \\mu$ be as in Definition \\ref{def:kakeya-model}. To fit our case, let us assume that for any $y \\in \\fJ$, $$\\cW(y) = \\tilde \\mu(\\Omega'(y)) \\le e^{-2 d_i s' } e^{-2s'}.$$ B", "anchor": { "source": "manuscript", "line_start": 2052, "line_end": 2052, "char_start": 114735, "char_end": 114757 }, "origin": "source_reference", "context": "statement" }, { "from": "lm:concentrated-surface", "to": "def:structure-randomness", "type": "internal_reference", "evidence": "for any $y \\in \\fJ$, $$\\cW(y) = \\tilde \\mu(\\Omega'(y)) \\le e^{-2 d_i s' } e^{-2s'}.$$ By Lemma \\ref{lm:concentrated-surface}, for every $y \\in \\fJ$ and $z \\in \\cI$ from the same cross-section $\\Sigma^r$, there is a corresponding neighborhood $\\cC(\\cT(y),", "anchor": { "source": "manuscript", "line_start": 2054, "line_end": 2054, "char_start": 114895, "char_end": 114924 }, "origin": "source_reference", "context": "statement" }, { "from": "lm:concentrated-surface", "to": "def:structure-randomness", "type": "internal_reference", "evidence": "$j$ be so that the size of the sides of $\\|(v_1, w_1)\\|$ are in $[e^{s'-j}, e^{s'-j+1}]$ (cf.~Lemma~\\ref{lm:concentrated-surface}, particularly equation \\eqref{eq:hyperbola-box}). Then it is easy to see that", "anchor": { "source": "manuscript", "line_start": 2057, "line_end": 2057, "char_start": 115335, "char_end": 115364 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:hyperbola-box", "to": "def:structure-randomness", "type": "internal_reference", "evidence": "\\|$ are in $[e^{s'-j}, e^{s'-j+1}]$ (cf.~Lemma~\\ref{lm:concentrated-surface}, particularly equation \\eqref{eq:hyperbola-box}). Then it is easy to see that \\fS(\\c", "anchor": { "source": "manuscript", "line_start": 2057, "line_end": 2057, "char_start": 115388, "char_end": 115412 }, "origin": "source_reference", "context": "statement" }, { "from": "subsec-constants", "to": "def:structure-randomness", "type": "internal_reference", "evidence": "e \\beta^{-201} e^{-2 d_i s' - j}, where $N_2$ is a large numeric constant (in \\S\\ref{subsec-constants},it was chosen to be $10^6$). The curve neighborhood $\\cC(\\cT(y), z)$ is called highly concentrated if $$ \\fS(\\cT(y),", "anchor": { "source": "manuscript", "line_start": 2062, "line_end": 2062, "char_start": 115723, "char_end": 115745 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:omega-y-r", "to": "prop:dimension-improvement-random", "type": "candidate_proof_dependency", "evidence": "lde z, r_{\\tilde{z}}) \\neq \\emptyset \\right\\}, \\] where $\\Omega(\\tilde z, r_{\\tilde{z}}) $ is as in \\eqref{eq:omega-y-r}. Then for every $\\hat z \\in \\fJ(\\tilde z, r)$, we have that $y, \\hat z$ satisfies $\\diamond_r$. Since \\[ \\Omega(\\tilde z, r_{\\ti", "anchor": { "source": "manuscript", "line_start": 2139, "line_end": 2139, "char_start": 120463, "char_end": 120483 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:assumption-1", "to": "prop:dimension-improvement-random", "type": "candidate_proof_dependency", "evidence": "s'} e^{-(d_i +\\epsilon_1)s'} \\\\ &= \\beta^{31} \\cW(y) e^{-(d_i +\\epsilon_1)s'}. By \\eqref{eq:assumption-1} and Lemma \\ref{lm:time-leave}, For every $ \\hat z \\in \\fJ\\cap\\Sigma^r$ such that $(y, \\hat z) \\in \\diamond_r$ and \\eqref{eq:ass", "anchor": { "source": "manuscript", "line_start": 2168, "line_end": 2168, "char_start": 122498, "char_end": 122521 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:time-leave", "to": "prop:dimension-improvement-random", "type": "candidate_proof_dependency", "evidence": "&= \\beta^{31} \\cW(y) e^{-(d_i +\\epsilon_1)s'}. By \\eqref{eq:assumption-1} and Lemma \\ref{lm:time-leave}, For every $ \\hat z \\in \\fJ\\cap\\Sigma^r$ such that $(y, \\hat z) \\in \\diamond_r$ and \\eqref{eq:assumption-1}, we have that $I(\\h", "anchor": { "source": "manuscript", "line_start": 2168, "line_end": 2168, "char_start": 122532, "char_end": 122551 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:assumption-1", "to": "prop:dimension-improvement-random", "type": "candidate_proof_dependency", "evidence": "m:time-leave}, For every $ \\hat z \\in \\fJ\\cap\\Sigma^r$ such that $(y, \\hat z) \\in \\diamond_r$ and \\eqref{eq:assumption-1}, we have that $I(\\hat y, \\hat z) \\subset I$ where $I$ is an interval of length $\\le e^{C_1 \\epsilon_1 s'}$. Note that the summand", "anchor": { "source": "manuscript", "line_start": 2169, "line_end": 2169, "char_start": 122638, "char_end": 122661 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:concentrated-surface", "to": "prop:dimension-improvement-random", "type": "candidate_proof_dependency", "evidence": "z'\\in \\cI_z$ such that there exists $r' \\in I(y, z')$ with $|r'| \\ge \\beta^{100} e^{2s'}$. By Lemma \\ref{lm:concentrated-surface}, $\\fD_{z}(z') \\in \\cC(\\cT(y), z)$. \\par Let us first consider the case $y \\in \\HS(z,r)$. By \\eqref{eq:fScT}, we have that \\beg", "anchor": { "source": "manuscript", "line_start": 2177, "line_end": 2177, "char_start": 123896, "char_end": 123925 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:fScT", "to": "prop:dimension-improvement-random", "type": "candidate_proof_dependency", "evidence": "face}, $\\fD_{z}(z') \\in \\cC(\\cT(y), z)$. \\par Let us first consider the case $y \\in \\HS(z,r)$. By \\eqref{eq:fScT}, we have that \\sum_{z' \\in \\cI_z, (y, z') \\in \\diamond_r} \\cW(y) \\cW(z') &\\le \\beta^{-101} \\cW(y) e^{2s'} e^", "anchor": { "source": "manuscript", "line_start": 2178, "line_end": 2178, "char_start": 124020, "char_end": 124035 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:cS-lower-bound", "to": "prop:dimension-improvement-random", "type": "candidate_proof_dependency", "evidence": "e^{-d_i s' - 6C_1\\epsilon_1 s'}. This upper bound is smaller than the lower bound in \\eqref{eq:cS-lower-bound} which leads to a contradiction. This proves that $\\tilde \\mu (\\bar \\cR^{\\fr}_i) < \\beta^{22}$. \\par Using arguments s", "anchor": { "source": "manuscript", "line_start": 2216, "line_end": 2217, "char_start": 126719, "char_end": 126744 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:dimension-improvement-random", "to": "sec-dimension-improvement-unstructured", "type": "internal_reference", "evidence": "ar \\cR^{\\fr}_i) < \\beta^{22}$. \\par Using arguments similar to the proof of Proposition \\ref{prop:dimension-improvement-random}, we will prove the measure of $\\cR_{i,j}$ is small: For any", "anchor": { "source": "manuscript", "line_start": 2219, "line_end": 2219, "char_start": 126909, "char_end": 126948 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:strange-nbhd-condition", "to": "prop:dimension-improvement-strange-part", "type": "candidate_proof_dependency", "evidence": "For each $\\Theta'(\\bar z) \\subset \\cR_{i,j}$, fix a subset $\\Xi_j(z) \\subset \\Theta'(\\bar z)$ as in \\eqref{eq:strange-nbhd-condition} and let $\\cR'_{i,j}$ denote the union of all of those $\\Xi_j(z)$ for $\\Theta'(\\bar z) \\subset \\cR_{i,j}$. To prove $\\tilde \\mu_{\\", "anchor": { "source": "manuscript", "line_start": 2226, "line_end": 2226, "char_start": 127298, "char_end": 127331 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:dimension-improvement-random", "to": "prop:dimension-improvement-strange-part", "type": "candidate_proof_dependency", "evidence": "de \\mu_{\\fr}(\\cR'_{i,j}) \\ge \\beta^{22.1}$. Using similar arguments as in the proof of Proposition \\ref{prop:dimension-improvement-random}, we can get a subset $\\cR''_{i,j} \\subset \\cR'_{i,j} $ such that $\\tilde \\mu_{\\fr}(\\cR''_{i,j}) \\ge \\beta^{22.2}$ and for any $y", "anchor": { "source": "manuscript", "line_start": 2229, "line_end": 2229, "char_start": 127704, "char_end": 127743 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:dimension-improvement-random", "to": "prop:dimension-improvement-strange-part", "type": "candidate_proof_dependency", "evidence": "cap(\\Xi_j(z') \\setminus \\Xi_{j+1}(y'))$. Following the same argument as in the proof of Proposition \\ref{prop:dimension-improvement-random}, for every $y'_2 \\in \\cI_{y'\\!, j}$ we can get a finite subset $\\fJ_j(y'_2, r) \\subset \\fJ\\cap\\Sigma^r$ such that for any $y_2", "anchor": { "source": "manuscript", "line_start": 2252, "line_end": 2252, "char_start": 129353, "char_end": 129392 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:dimension-improvement-random", "to": "prop:dimension-improvement-strange-part", "type": "candidate_proof_dependency", "evidence": "d{align*} Similarly to the proof of the lower bound for $\\cS(\\Sigma^r)$ in the proof of Proposition \\ref{prop:dimension-improvement-random} (specifically, the discussion in p.~\\pageref{eq:assumption-1}), since \\[ |u(\\fl(\\hat y))\\hat y \\cap \\cR'_{i,j}| \\ge \\beta^{30} e", "anchor": { "source": "manuscript", "line_start": 2263, "line_end": 2263, "char_start": 130178, "char_end": 130217 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:time-leave", "to": "prop:dimension-improvement-strange-part", "type": "candidate_proof_dependency", "evidence": "2\\epsilon_1)s' }.\\nonumber Since $\\|\\Delta(y, y_2)\\| \\ge \\beta e^{-s' - j-1}$, by Lemma \\ref{lm:time-leave}, for each $y_2 \\in \\fJ\\cap \\Sigma^r$ such that $(y, y_2) \\in \\diamond_{r,j}$, we have that $I(y, y_2) \\subset I$ where $I$ is an", "anchor": { "source": "manuscript", "line_start": 2272, "line_end": 2272, "char_start": 130977, "char_end": 130996 }, "origin": "source_reference", "context": "proof" }, { "from": "eq: triple sum", "to": "prop:dimension-improvement-strange-part", "type": "candidate_proof_dependency", "evidence": "interval of length $\\le e^{j+1}$. Note that the summand $\\cW(y) \\cW(y_2)$ appears in the summation \\eqref{eq: triple sum} for $y' \\in \\cY_j$ only if $y' = u(r_{y'}) \\hat y$ where $r_{y'} \\in I(y, y_2)$. Therefore, every summand $\\cW(y) \\cW(y_2)$ appea", "anchor": { "source": "manuscript", "line_start": 2272, "line_end": 2272, "char_start": 131226, "char_end": 131248 }, "origin": "source_reference", "context": "proof" }, { "from": "eq: triple sum", "to": "prop:dimension-improvement-strange-part", "type": "candidate_proof_dependency", "evidence": "r_{y'}) \\hat y$ where $r_{y'} \\in I(y, y_2)$. Therefore, every summand $\\cW(y) \\cW(y_2)$ appears in \\eqref{eq: triple sum} for at most $\\beta^{-1} e^{j+1}$ different $y' \\in \\cY_j$. This implies that for a fixed $y \\in \\fJ \\cap \\Sigma^r\\cap \\cR''_{i,j", "anchor": { "source": "manuscript", "line_start": 2272, "line_end": 2272, "char_start": 131384, "char_end": 131406 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:concentrated-surface", "to": "lem:sigma j lemma", "type": "candidate_proof_dependency", "evidence": "\\vv r^-$ satisfies that $\\|\\vv w^-\\| \\in [\\beta e^{-s' - j-1}, \\beta e^{-s' - j }]$. Then by Lemma~\\ref{lm:concentrated-surface}, we have $\\|\\vv w^+\\| \\le \\beta e^{s' - j}$. Let $\\hat y_1 = a(-2t) y_1, \\hat y_2 = a(-2t) y_2$. Then we have \\[ \\hat y_1 = a_0 a", "anchor": { "source": "manuscript", "line_start": 2303, "line_end": 2303, "char_start": 133236, "char_end": 133265 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:product-w", "to": "lem:sigma j lemma", "type": "candidate_proof_dependency", "evidence": "v w}^- \\in B_{\\vv r^-}(\\beta e^{-j})$, and $\\hat{\\vv w}^+ \\in B_{\\vv r^+}(\\beta e^{-j})$. By Lemma \\ref{lm:product-w}, \\[ \\exp (\\hat{ \\vv w}^-)\\exp (\\hat{ \\vv w}^+) = \\exp(\\hat{\\vv w})\\exp(\\hat{\\vv h}), \\] where $\\hat{\\vv w} \\in B_{\\fr_1 + \\fr_2}", "anchor": { "source": "manuscript", "line_start": 2305, "line_end": 2305, "char_start": 133631, "char_end": 133649 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:hat-y1-y2", "to": "lem:sigma j lemma", "type": "candidate_proof_dependency", "evidence": "\\vv w}_1 \\in B_{\\fr_1 + \\fr_2}(\\beta e^{-j})$, and $\\tilde{\\vv h}_1 \\in B_{\\R \\fa_0 + \\fh}(\\beta)$. \\eqref{eq:hat-y1-y2} and \\eqref{eq:hat-y1-z} imply that \\[ \\hat y_2 = \\exp(-\\tilde{\\vv h}) \\exp(-\\tilde{\\vv w})\\exp(\\tilde{\\vv w}_1) \\exp(\\tilde{\\vv", "anchor": { "source": "manuscript", "line_start": 2319, "line_end": 2319, "char_start": 134632, "char_end": 134652 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:hat-y1-z", "to": "lem:sigma j lemma", "type": "candidate_proof_dependency", "evidence": "fr_2}(\\beta e^{-j})$, and $\\tilde{\\vv h}_1 \\in B_{\\R \\fa_0 + \\fh}(\\beta)$. \\eqref{eq:hat-y1-y2} and \\eqref{eq:hat-y1-z} imply that \\[ \\hat y_2 = \\exp(-\\tilde{\\vv h}) \\exp(-\\tilde{\\vv w})\\exp(\\tilde{\\vv w}_1) \\exp(\\tilde{\\vv h}_1) \\hat z'. \\] By Le", "anchor": { "source": "manuscript", "line_start": 2319, "line_end": 2319, "char_start": 134657, "char_end": 134676 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:product-w", "to": "lem:sigma j lemma", "type": "candidate_proof_dependency", "evidence": "tilde{\\vv h}) \\exp(-\\tilde{\\vv w})\\exp(\\tilde{\\vv w}_1) \\exp(\\tilde{\\vv h}_1) \\hat z'. \\] By Lemma \\ref{lm:product-w}, we have \\[ \\hat y_2 = \\exp(\\tilde{\\vv w}_2) \\exp(\\tilde{\\vv h}_2) \\hat z', \\] where $\\tilde{\\vv w}_2 \\in B_{\\fr_1 + \\fr_2}(2.1", "anchor": { "source": "manuscript", "line_start": 2321, "line_end": 2321, "char_start": 134810, "char_end": 134828 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:sigma j lemma", "to": "prop:dimension-improvement-strange-part", "type": "candidate_proof_dependency", "evidence": "') \\subset \\Sigma^r \\cap \\cR'_{i,j} $. Using similar arguments to those used in the proof of Lemma~\\ref{lem:sigma j lemma}, we can prove that \\[ \\Xi_j(z') \\subset \\Sigma_{s', j-2}(y_2). \\] Define \\cI(y", "anchor": { "source": "manuscript", "line_start": 2330, "line_end": 2330, "char_start": 135487, "char_end": 135510 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:concentrated-surface", "to": "prop:dimension-improvement-strange-part", "type": "candidate_proof_dependency", "evidence": ", j-2}(y_2) \\cap \\cR'_{i,j} \\Bigr\\}. Fix $z \\in \\cI(y_2, j)$. By Lemma \\ref{lm:concentrated-surface}, the coordinates $\\fD_{ z}(y_1)$ of all $y_1\\in \\cI_z$ such that $(y_2,y_1) \\in \\diamond_{r,j}$ are contained in $$\\cC(\\cT(y_2)", "anchor": { "source": "manuscript", "line_start": 2339, "line_end": 2339, "char_start": 135908, "char_end": 135937 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:concentrated-surface", "to": "prop:dimension-improvement-strange-part", "type": "candidate_proof_dependency", "evidence": "j}$ are contained in $$\\cC(\\cT(y_2), z) \\cap \\fB(\\cT(y_2), z).$$ By $\\diamond_{r,j}$ and Lemma \\ref{lm:concentrated-surface}, we have that if there is at least one such $y_1$ the size of $\\fB(\\cT(y_2), z)$ is $\\le \\beta^{-98} e^{-s'-j}$.", "anchor": { "source": "manuscript", "line_start": 2343, "line_end": 2343, "char_start": 136130, "char_end": 136159 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:fScT", "to": "prop:dimension-improvement-strange-part", "type": "candidate_proof_dependency", "evidence": "Using \\eqref{eq:fScT} we may conclude that \\[ \\sum_{\\fD_{ z}(y_1) \\in \\cC(\\cT(y_2), z)\\cap \\fB(\\cT(y_2), z)} \\cW(y_1)\\cW(y_2) \\ll \\cW(y_2) \\beta^{-", "anchor": { "source": "manuscript", "line_start": 2347, "line_end": 2347, "char_start": 136389, "char_end": 136404 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:initial-dimension-under-a-t", "to": "prop:dimension-improvement-strange-part", "type": "candidate_proof_dependency", "evidence": "cap \\fB(\\cT(y_2), z)} \\cW(y_1)\\cW(y_2) \\ll \\cW(y_2) \\beta^{-198} e^{-j} e^{-2d_i s'}. \\] By Lemma \\ref{lm:initial-dimension-under-a-t}, we have \\[ \\mu(\\Sigma_{s', j-2}(y_2)) \\le e^{-2s'} e^{- d_0 (j-2)}.\\] Since $\\Pi(x)$ is a full playground, we have \\[ \\tilde \\", "anchor": { "source": "manuscript", "line_start": 2349, "line_end": 2349, "char_start": 136572, "char_end": 136608 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:cIy2j", "to": "prop:dimension-improvement-strange-part", "type": "candidate_proof_dependency", "evidence": "de \\mu(\\Sigma_{s', j-2}(y_2)) \\le \\beta^{-15}e^{-2s'} e^{- d_0 (j-2)}.\\] Noting that for $z'$ as in \\eqref{eq:cIy2j} \\[ \\tilde \\mu_{\\fr}(\\Xi_j(z')) \\ge (1-\\beta)^j e^{-2s'} e^{-(d_i + \\epsilon_1)s'}, \\] we have that |\\cI(y_2, j)|", "anchor": { "source": "manuscript", "line_start": 2353, "line_end": 2353, "char_start": 136834, "char_end": 136850 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:j lower bound", "to": "prop:dimension-improvement-strange-part", "type": "candidate_proof_dependency", "evidence": "\\epsilon_1 s'} e^{- d_0 j} e^{-d_i s' - j}. \\] This upper bound is smaller than the lower bound in \\eqref{eq:j lower bound} because $ d_0 j \\ge d_0 C_1 \\epsilon_1 s' \\ge 10 \\epsilon_1 s'$. This leads to a contradiction, completing the proof of Propositi", "anchor": { "source": "manuscript", "line_start": 2370, "line_end": 2370, "char_start": 137896, "char_end": 137920 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:dimension-improvement-random", "to": "sec-no-loss-of-dimension", "type": "internal_reference", "evidence": "\\par Using similar arguments to those in the proofs of Proposition \\ref{prop:dimension-improvement-random} and Proposition \\ref{prop:dimension-improvement-strange-part}, we can also prove that compared to $\\mu_i$, the dimension of $\\til", "anchor": { "source": "manuscript", "line_start": 2380, "line_end": 2380, "char_start": 138832, "char_end": 138871 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:dimension-improvement-strange-part", "to": "sec-no-loss-of-dimension", "type": "internal_reference", "evidence": "ments to those in the proofs of Proposition \\ref{prop:dimension-improvement-random} and Proposition \\ref{prop:dimension-improvement-strange-part}, we can also prove that compared to $\\mu_i$, the dimension of $\\tilde \\mu$ does not drop significantly: \\label{pro", "anchor": { "source": "manuscript", "line_start": 2380, "line_end": 2380, "char_start": 138888, "char_end": 138933 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:improve-dimension", "to": "prop:dimension-does-not-drop", "type": "internal_reference", "evidence": "y: Let us keep the notation as in Proposition \\ref{prop:improve-dimension}. Then there exists $\\mathfrak{Z} \\subset \\Pi(x)$ with $\\tilde \\mu(\\mathfrak Z) \\le \\beta^{50}$ such that for $x \\in \\cI\\setminus", "anchor": { "source": "manuscript", "line_start": 2383, "line_end": 2383, "char_start": 139135, "char_end": 139163 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:strange-normal-decomposition", "to": "prop:dimension-does-not-drop", "type": "candidate_proof_dependency", "evidence": "(z)$'s. Then it suffices to show that \\[ \\tilde \\mu (\\mathfrak Z ) \\le \\beta^{50}. \\] Similarly to \\eqref{eq:strange-normal-decomposition}, let us decompose \\[ \\fZ = \\fZ' \\cup \\bigcup_{ C_1 \\epsilon_1 s' \\le j \\le 2s'} \\fZ_j. \\] Here $\\fZ'$ denotes the union of extrem", "anchor": { "source": "manuscript", "line_start": 2394, "line_end": 2394, "char_start": 139845, "char_end": 139884 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:dimension-improvement-random", "to": "prop:dimension-does-not-drop", "type": "candidate_proof_dependency", "evidence": "sketch the proof of $$\\tilde \\mu(\\fZ') \\le \\beta^{50.1},$$ which follows the proof of Proposition \\ref{prop:dimension-improvement-random}. \\par Assume in contradiction that \\[ \\tilde \\mu(\\fZ') > \\beta^{50.1}. \\] By repeating the argument in the proof of Proposition", "anchor": { "source": "manuscript", "line_start": 2410, "line_end": 2410, "char_start": 140791, "char_end": 140830 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:dimension-improvement-random", "to": "prop:dimension-does-not-drop", "type": "candidate_proof_dependency", "evidence": "that \\[ \\tilde \\mu(\\fZ') > \\beta^{50.1}. \\] By repeating the argument in the proof of Proposition \\ref{prop:dimension-improvement-random}, we get $\\fZ'' \\subset \\fZ'$ and a cross-section $\\Sigma^r$ such that \\item $\\tilde \\mu(\\fZ'') \\ge \\beta^{5", "anchor": { "source": "manuscript", "line_start": 2413, "line_end": 2413, "char_start": 140960, "char_end": 140999 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:dimension-improvement-random", "to": "prop:dimension-does-not-drop", "type": "candidate_proof_dependency", "evidence": ") \\cap \\fZ') \\neq \\emptyset$. We can repeat the argument in the proof of Proposition \\ref{prop:dimension-improvement-random} to get lower and upper bounds on $\\cZ(\\Sigma^r)$ (similarly to $\\cS(\\Sigma^r)$): \\par For lower bound, we get that for any $y_1", "anchor": { "source": "manuscript", "line_start": 2428, "line_end": 2428, "char_start": 141796, "char_end": 141835 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:theta sum'", "to": "prop:dimension-does-not-drop", "type": "candidate_proof_dependency", "evidence": "in \\Omega(y_2)$ so $u(r') y_2' \\in \\Theta'(z)$. However each such $(y_1,y_2)$ may appear in the sum \\eqref{eq:theta sum'} up to $|I(y_1,y_2)|\\beta^{-1}$ many times. Throwing away all $y_2$ in for which the corresponding $u(r')$ trajectory meets $\\Thet", "anchor": { "source": "manuscript", "line_start": 2440, "line_end": 2440, "char_start": 142833, "char_end": 142854 }, "origin": "source_reference", "context": "proof" }, { "from": "eq: y_2 sum for diamond'", "to": "prop:dimension-does-not-drop", "type": "candidate_proof_dependency", "evidence": "ood we can ensure that $|I(y_1,y_2)|\\lessapprox e^{C_1\\epsilon_1s'}$. Combining everything one gets \\eqref{eq: y_2 sum for diamond'}. By summing up all $y_1 \\in \\fJ \\cap \\Sigma^r$ with $\\Omega'(y_1) \\cap \\fZ'' \\neq \\emptyset$, we have \\cZ(\\Sigma", "anchor": { "source": "manuscript", "line_start": 2441, "line_end": 2441, "char_start": 143139, "char_end": 143171 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:concentrated-surface", "to": "prop:dimension-does-not-drop", "type": "candidate_proof_dependency", "evidence": "_1 s'}, where $\\tilde \\fm = \\tilde \\mu (\\Sigma^r)$. \\par For the upper bound, by Lemma \\ref{lm:concentrated-surface}, for any $y_1 \\in \\fJ \\cap \\Sigma^r$ and $z \\in \\cI \\cap \\Sigma^r \\cap \\fZ'$, we have that for any $y_2 \\in \\cI_z$ such that $(y_", "anchor": { "source": "manuscript", "line_start": 2448, "line_end": 2448, "char_start": 143596, "char_end": 143625 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:fScT", "to": "prop:dimension-does-not-drop", "type": "candidate_proof_dependency", "evidence": "\\cI_z$ such that $(y_1, y_2) \\in \\diamond'_r$, we have \\[ \\fD_z (y_2) \\in \\cC(\\cT(y_1), z). \\] By \\eqref{eq:fScT}, we get \\sum_{y_2 \\in \\cI_z: (y_1, y_2) \\in \\diamond'_r} \\cW(y_1) \\cW(y_2) &\\le \\cW(y_1) \\beta^{-101} e^{-2d_", "anchor": { "source": "manuscript", "line_start": 2450, "line_end": 2450, "char_start": 143833, "char_end": 143848 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:dimension-improvement-strange-part", "to": "prop:dimension-does-not-drop", "type": "candidate_proof_dependency", "evidence": "50.1}, \\text{ for any } j \\in [C_1 \\epsilon_1 s', 2s'], \\] which follows the proof of Proposition \\ref{prop:dimension-improvement-strange-part}. \\par For every $z \\in \\cI \\cap \\Sigma^r \\cap \\fZ_j$, let us fix a $\\Xi_j(z') \\subset \\Theta'(z)$ such that \\[ \\tilde \\mu (\\Xi", "anchor": { "source": "manuscript", "line_start": 2469, "line_end": 2469, "char_start": 145005, "char_end": 145050 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:dimension-improvement-random", "to": "prop:dimension-does-not-drop", "type": "candidate_proof_dependency", "evidence": "at \\[\\tilde \\mu(\\fZ'_j) > \\beta^{50.1}. \\] Repeating the argument as in the proof of Proposition \\ref{prop:dimension-improvement-random}, we will get $\\fZ''_j \\subset \\fZ'_j$ and a cross-section $\\Sigma^r$ satisfying \\item $\\tilde \\mu (\\fZ''_j)", "anchor": { "source": "manuscript", "line_start": 2478, "line_end": 2478, "char_start": 145635, "char_end": 145674 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:dimension-improvement-strange-part", "to": "prop:dimension-does-not-drop", "type": "candidate_proof_dependency", "evidence": ") \\cap \\fZ') \\neq \\emptyset$. We can repeat the argument in the proof of Proposition \\ref{prop:dimension-improvement-strange-part} to get lower and upper bounds on $\\cZ_j(\\Sigma^r)$ (similarly to $\\cS_j(\\Sigma^r)$): \\par For lower bound, we will get that for a", "anchor": { "source": "manuscript", "line_start": 2493, "line_end": 2493, "char_start": 146583, "char_end": 146628 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:concentrated-surface", "to": "prop:dimension-does-not-drop", "type": "candidate_proof_dependency", "evidence": "p \\fZ'_j: (y_1, y_2) \\in \\diamond'_{r,j} \\right\\}. \\] Then for every $z \\in \\cI'(y_1, j)$, by Lemma \\ref{lm:concentrated-surface}, for any $y_2 \\in \\fJ(y_1, z, j)$, we have \\[ \\fD_z(y_2) \\in \\cC(\\cT(y_1), z) \\cap \\fB(\\cT(y_1), z). \\] Therefore, \\[ \\sum_{y_2", "anchor": { "source": "manuscript", "line_start": 2522, "line_end": 2522, "char_start": 148253, "char_end": 148282 }, "origin": "source_reference", "context": "proof" }, { "from": "def:structure-random-decompostion", "to": "sec-structured-component", "type": "internal_reference", "evidence": "ement --- structured part} \\par Let us keep notation in Definition \\ref{def:structure-random-decompostion}. This section is devoted to the study of the structured part $\\bar \\cR^{\\fs}_i$. \\par We will prove the following statement:", "anchor": { "source": "manuscript", "line_start": 2555, "line_end": 2555, "char_start": 150119, "char_end": 150158 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:strange-normal-decomposition", "to": "sec-structured-component", "type": "internal_reference", "evidence": "1 $. We have \\[ \\tilde \\mu (\\bar \\cR^{\\fs}_i) \\le \\beta^{22}. \\] \\par Note that by \\eqref{eq:strange-normal-decomposition} and \\eqref{eq:normal-random-structure-decomposition}, Proposition \\ref{prop:dimension-improvement-random}, \\ref{prop:dimension-im", "anchor": { "source": "manuscript", "line_start": 2567, "line_end": 2567, "char_start": 150505, "char_end": 150544 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:normal-random-structure-decomposition", "to": "sec-structured-component", "type": "internal_reference", "evidence": "\\fs}_i) \\le \\beta^{22}. \\] \\par Note that by \\eqref{eq:strange-normal-decomposition} and \\eqref{eq:normal-random-structure-decomposition}, Proposition \\ref{prop:dimension-improvement-random}, \\ref{prop:dimension-improvement-strange-part} and \\ref{prop:dimension-impro", "anchor": { "source": "manuscript", "line_start": 2567, "line_end": 2567, "char_start": 150549, "char_end": 150597 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:dimension-improvement-random", "to": "sec-structured-component", "type": "internal_reference", "evidence": "{eq:strange-normal-decomposition} and \\eqref{eq:normal-random-structure-decomposition}, Proposition \\ref{prop:dimension-improvement-random}, \\ref{prop:dimension-improvement-strange-part} and \\ref{prop:dimension-improvement-structured-component} will imply \\eqref{eq:hea", "anchor": { "source": "manuscript", "line_start": 2567, "line_end": 2567, "char_start": 150611, "char_end": 150650 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:dimension-improvement-strange-part", "to": "sec-structured-component", "type": "internal_reference", "evidence": "ref{eq:normal-random-structure-decomposition}, Proposition \\ref{prop:dimension-improvement-random}, \\ref{prop:dimension-improvement-strange-part} and \\ref{prop:dimension-improvement-structured-component} will imply \\eqref{eq:heavy-part-is-small} and thus Proposition \\ref{pro", "anchor": { "source": "manuscript", "line_start": 2567, "line_end": 2567, "char_start": 150652, "char_end": 150697 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:dimension-improvement-structured-component", "to": "sec-structured-component", "type": "internal_reference", "evidence": "position \\ref{prop:dimension-improvement-random}, \\ref{prop:dimension-improvement-strange-part} and \\ref{prop:dimension-improvement-structured-component} will imply \\eqref{eq:heavy-part-is-small} and thus Proposition \\ref{prop:improve-dimension}. \\par Note that up to now, from the", "anchor": { "source": "manuscript", "line_start": 2567, "line_end": 2567, "char_start": 150702, "char_end": 150755 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:heavy-part-is-small", "to": "sec-structured-component", "type": "internal_reference", "evidence": "sion-improvement-strange-part} and \\ref{prop:dimension-improvement-structured-component} will imply \\eqref{eq:heavy-part-is-small} and thus Proposition \\ref{prop:improve-dimension}. \\par Note that up to now, from the results of \\S\\ref{subsec-initialize} we ha", "anchor": { "source": "manuscript", "line_start": 2567, "line_end": 2567, "char_start": 150767, "char_end": 150797 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:improve-dimension", "to": "sec-structured-component", "type": "internal_reference", "evidence": "on-improvement-structured-component} will imply \\eqref{eq:heavy-part-is-small} and thus Proposition \\ref{prop:improve-dimension}. \\par Note that up to now, from the results of \\S\\ref{subsec-initialize} we have only used Proposition~\\ref{prop:initial-bound-d", "anchor": { "source": "manuscript", "line_start": 2567, "line_end": 2567, "char_start": 150819, "char_end": 150847 }, "origin": "source_reference", "context": "narrative" }, { "from": "subsec-initialize", "to": "sec-structured-component", "type": "internal_reference", "evidence": "and thus Proposition \\ref{prop:improve-dimension}. \\par Note that up to now, from the results of \\S\\ref{subsec-initialize} we have only used Proposition~\\ref{prop:initial-bound-dimension}. Proposition~\\ref{prop:dimension-improvement-structured-componen", "anchor": { "source": "manuscript", "line_start": 2569, "line_end": 2569, "char_start": 150898, "char_end": 150921 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:initial-bound-dimension", "to": "sec-structured-component", "type": "internal_reference", "evidence": "ar Note that up to now, from the results of \\S\\ref{subsec-initialize} we have only used Proposition~\\ref{prop:initial-bound-dimension}. Proposition~\\ref{prop:dimension-improvement-structured-component} uses in an essential way Proposition~\\ref{prop:sl2xr2-closing-", "anchor": { "source": "manuscript", "line_start": 2569, "line_end": 2569, "char_start": 150952, "char_end": 150986 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:dimension-improvement-structured-component", "to": "sec-structured-component", "type": "internal_reference", "evidence": "ef{subsec-initialize} we have only used Proposition~\\ref{prop:initial-bound-dimension}. Proposition~\\ref{prop:dimension-improvement-structured-component} uses in an essential way Proposition~\\ref{prop:sl2xr2-closing-lemma}, which essentially says $\\tilde \\mu$ cannot concentrate alon", "anchor": { "source": "manuscript", "line_start": 2569, "line_end": 2569, "char_start": 151000, "char_end": 151053 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:sl2xr2-closing-lemma", "to": "sec-structured-component", "type": "internal_reference", "evidence": "position~\\ref{prop:dimension-improvement-structured-component} uses in an essential way Proposition~\\ref{prop:sl2xr2-closing-lemma}, which essentially says $\\tilde \\mu$ cannot concentrate along $\\SL(2,\\R)\\ltimes\\R^2$-orbits. \\par Let us first fix some notation", "anchor": { "source": "manuscript", "line_start": 2569, "line_end": 2569, "char_start": 151091, "char_end": 151122 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:w1w'1", "to": "lm:hyperbola-plane", "type": "internal_reference", "evidence": "e Ae^{-s'+2\\ell}, \\text{ and } |k - k'| \\le Ae^{-2s' +2\\ell}. Conversely, if \\eqref{eq:w1w'1} holds, then $C'$ is contained in the $A e^{-3s'}$ neighborhood of $C$. \\par (2) Suppose $C'$ is not contained in the $e^{-3s", "anchor": { "source": "manuscript", "line_start": 2599, "line_end": 2599, "char_start": 152745, "char_end": 152761 }, "origin": "source_reference", "context": "statement" }, { "from": "lm:hyperbola-plane", "to": "sec-structured-component", "type": "internal_reference", "evidence": "; \\node at (7.5,9.7) { $\\vv p'$}; \\node at (-0.6,0.5) { $(0,0)$}; \\caption{Lemma \\ref{lm:hyperbola-plane}, part (2) --- if there is a $\\vv p' \\in C'$, not too close to the origin, near $C$ then all of $C'$ is within a comparable distan", "anchor": { "source": "manuscript", "line_start": 2619, "line_end": 2619, "char_start": 154276, "char_end": 154300 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:w1w'1", "to": "sec-structured-component", "type": "internal_reference", "evidence": "ll}. \\] Moreover \\[ |k - k'| < e^{-s'}|w_{1} - w'_{1}| \\le A e^{-2s'+2\\ell},\\] establishing \\eqref{eq:w1w'1}. For Subcase 2, we have \\[ |k - k'| e^{-s' - 2 \\ell} \\le Ae^{-3s'}, \\] implying \\[ |k - k'| \\le A e^{-2s'+2\\e", "anchor": { "source": "manuscript", "line_start": 2638, "line_end": 2638, "char_start": 155302, "char_end": 155318 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:w1w'1", "to": "sec-structured-component", "type": "internal_reference", "evidence": "-2s'+2\\ell}. \\] Then \\[ |w_{1} - w'_{1}| \\le e^{s'} |k - k'| \\le Ae^{-s'+2\\ell}. \\] This proves \\eqref{eq:w1w'1} and hence the first part of (1). \\par Now let us assume \\eqref{eq:w1w'1}, we want to prove that for any $Y \\in [\\beta e^{-s'}]$,", "anchor": { "source": "manuscript", "line_start": 2647, "line_end": 2647, "char_start": 155540, "char_end": 155556 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:w1w'1", "to": "sec-structured-component", "type": "internal_reference", "evidence": "'+2\\ell}. \\] This proves \\eqref{eq:w1w'1} and hence the first part of (1). \\par Now let us assume \\eqref{eq:w1w'1}, we want to prove that for any $Y \\in [\\beta e^{-s'}]$, \\[ \\left|\\frac{Y}{w_{1} Y + k} - \\frac{Y}{w'_{1}Y + k'} \\right| \\le", "anchor": { "source": "manuscript", "line_start": 2648, "line_end": 2648, "char_start": 155614, "char_end": 155630 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:y-y1-lhs", "to": "sec-structured-component", "type": "internal_reference", "evidence": "ght| &\\le |k - k'| \\le A e^{-2s' + 2\\ell}. This shows that for both subcases we have \\eqref{eq:y-y1-lhs} is bounded from above by $A e^{-3s'}$, which implies the second statement of (1). \\par Let us prove (2). By the assumptions, we", "anchor": { "source": "manuscript", "line_start": 2664, "line_end": 2664, "char_start": 156667, "char_end": 156686 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:k over w claim", "to": "sec-structured-component", "type": "internal_reference", "evidence": "11 C_1 \\epsilon_1 s'}$ neighborhood of $C$, which contradicts our assumption. This proves the claim~\\eqref{eq:k over w claim}. \\par From \\eqref{eq:w1-w1'} and \\eqref{eq:k over w claim} we obtain that for any $Y \\in [\\beta e^{-s'}]$ \\[ |Y| \\cdot |w_{1}", "anchor": { "source": "manuscript", "line_start": 2687, "line_end": 2687, "char_start": 158317, "char_end": 158342 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:w1-w1'", "to": "sec-structured-component", "type": "internal_reference", "evidence": "$C$, which contradicts our assumption. This proves the claim~\\eqref{eq:k over w claim}. \\par From \\eqref{eq:w1-w1'} and \\eqref{eq:k over w claim} we obtain that for any $Y \\in [\\beta e^{-s'}]$ \\[ |Y| \\cdot |w_{1} - w'_{1}| \\frac{\\left| Y + \\fr", "anchor": { "source": "manuscript", "line_start": 2688, "line_end": 2688, "char_start": 158356, "char_end": 158373 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:k over w claim", "to": "sec-structured-component", "type": "internal_reference", "evidence": "our assumption. This proves the claim~\\eqref{eq:k over w claim}. \\par From \\eqref{eq:w1-w1'} and \\eqref{eq:k over w claim} we obtain that for any $Y \\in [\\beta e^{-s'}]$ \\[ |Y| \\cdot |w_{1} - w'_{1}| \\frac{\\left| Y + \\frac{k' - k}{w_{1} - w'_{1}} \\ri", "anchor": { "source": "manuscript", "line_start": 2688, "line_end": 2688, "char_start": 158378, "char_end": 158403 }, "origin": "source_reference", "context": "narrative" }, { "from": "lm:hyperbola-plane", "to": "sec-structured-component", "type": "internal_reference", "evidence": "{-s' + 11 C_1 \\epsilon_1 s'}. \\] This completes the proof. \\par Let us apply Lemma \\ref{lm:hyperbola-plane} to prove the following lemmata: For $y \\in \\fJ\\cap \\Sigma^r , z \\in \\cI\\cap\\Sigma", "anchor": { "source": "manuscript", "line_start": 2703, "line_end": 2703, "char_start": 159366, "char_end": 159390 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:prime curve", "to": "lm:integrable-surface", "type": "candidate_proof_dependency", "evidence": "^\\ell, e^{\\ell +1}]$ and $ k' \\in [e^{\\ell-1}, e^{\\ell +2}]$. Note that the equations of the curves~\\eqref{eq:prime curve} and~\\eqref{eq:not prime curve} can be written as \\[ X = \\frac{-Y}{w_{1,y} Y + k}, \\text{ and } X = \\frac{-Y}{w_{1,y'}Y + k'} \\", "anchor": { "source": "manuscript", "line_start": 2747, "line_end": 2747, "char_start": 162184, "char_end": 162206 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:not prime curve", "to": "lm:integrable-surface", "type": "candidate_proof_dependency", "evidence": "k' \\in [e^{\\ell-1}, e^{\\ell +2}]$. Note that the equations of the curves~\\eqref{eq:prime curve} and~\\eqref{eq:not prime curve} can be written as \\[ X = \\frac{-Y}{w_{1,y} Y + k}, \\text{ and } X = \\frac{-Y}{w_{1,y'}Y + k'} \\] respectively. Then we hav", "anchor": { "source": "manuscript", "line_start": 2747, "line_end": 2747, "char_start": 162211, "char_end": 162237 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:hyperbola-plane", "to": "lm:integrable-surface", "type": "candidate_proof_dependency", "evidence": "{w_{1,y} Y + k} - \\frac{Y}{w_{1,y'}Y + k'} \\right| \\le 2e^{-3s'+11C_1\\epsilon_1 s'}.\\] By Lemma \\ref{lm:hyperbola-plane}(1) with $A =2 e^{11C_1\\epsilon_1 s'}$, we have that \\[ |w_{1,y} - w_{1, y'}| \\le 2e^{-s' + 11C_1 \\epsilon_1 s'+2\\ell}, \\text{", "anchor": { "source": "manuscript", "line_start": 2752, "line_end": 2752, "char_start": 162521, "char_end": 162545 }, "origin": "source_reference", "context": "proof" }, { "from": "rmk:concentrated-surface", "to": "lm:integrable-surface", "type": "candidate_proof_dependency", "evidence": "statement for $\\ell < s'$. \\par Now let us turn to the case $\\ell \\ge s'$. In this case, by Remark \\ref{rmk:concentrated-surface}, we can consider the corresponding straight lines \\[ X = -Y/k, \\text{ and } X = - Y/k'. \\] Then we have for any $y \\in [\\beta e^", "anchor": { "source": "manuscript", "line_start": 2760, "line_end": 2760, "char_start": 163232, "char_end": 163262 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:hyperbola-plane", "to": "lm:determine-curve-nbhd", "type": "candidate_proof_dependency", "evidence": "Y_1}{w_1 Y_1 + k} - \\frac{Y_1}{w'_{1}Y_1 + k'} \\right| \\le \\beta^{-100}e^{-3s'}. \\] By Lemma \\ref{lm:hyperbola-plane}(2) with $A = e^{2s' - 2\\ell -m}$ we have that $\\cC'$ is contained in the $e^{-s' - m - 2\\ell}$-neighborhood of $\\cC$. Recalling t", "anchor": { "source": "manuscript", "line_start": 2802, "line_end": 2802, "char_start": 165930, "char_end": 165954 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:hyperbola-plane", "to": "lm:counting-curves-nbhd", "type": "candidate_proof_dependency", "evidence": "\\[ \\left|\\frac{Y}{w_1 Y + k} - \\frac{Y}{w'_{1}Y + k'} \\right| \\le e^{-s' - m - 2\\ell}.\\] By Lemma \\ref{lm:hyperbola-plane} (1) with $A = e^{2s' - m - 2 \\ell}$, we have |w_1 - w'_{1}| \\le e^{s' -m }, \\te", "anchor": { "source": "manuscript", "line_start": 2832, "line_end": 2832, "char_start": 167991, "char_end": 168015 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:hyperbola-plane", "to": "lm:counting-curves-nbhd", "type": "candidate_proof_dependency", "evidence": "1 - w'_{1}| \\le e^{s' -m }, \\text{ and } |k - k'| \\le e^{-m }. Note that by Lemma \\ref{lm:hyperbola-plane} (1) with $A = e^{11C_1 \\epsilon_1 s'}$, if $$|w'_1 - \\bar w'_1| \\le e^{-s'+ 11C_1 \\epsilon_1 s'+2\\ell}$$ and $$|k' - \\bar k'|", "anchor": { "source": "manuscript", "line_start": 2838, "line_end": 2838, "char_start": 168213, "char_end": 168237 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:counting-conditions", "to": "lm:counting-curves-nbhd", "type": "candidate_proof_dependency", "evidence": "_1 Y + k'} \\] and \\[ X = \\frac{-Y}{\\bar w'_1 Y + \\bar k'} \\] are comparable. Thus, the conditions \\eqref{eq:counting-conditions} give at most \\[ e^{4s' - 2m - 4\\ell} \\] different curve neighborhoods, which are not comparable to each other. \\par Let us prov", "anchor": { "source": "manuscript", "line_start": 2847, "line_end": 2847, "char_start": 168557, "char_end": 168587 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:hyperbola-plane", "to": "lm:counting-curves-nbhd", "type": "candidate_proof_dependency", "evidence": "ghborhoods, which are not comparable to each other. \\par Let us prove the last statement. By Lemma \\ref{lm:hyperbola-plane} (2) with $A = e^{2s' -2\\ell -m'}$, we have that there exists $(X_1, Y_1) \\in \\cC$ such that the intersection \\[\\cC\\cap \\cC' \\cap", "anchor": { "source": "manuscript", "line_start": 2850, "line_end": 2850, "char_start": 168748, "char_end": 168772 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:def-thin", "to": "sec-structured-component", "type": "internal_reference", "evidence": "(2s')Q(\\fr_1, \\ell-1, \\beta ) B^{A_0 H}(\\beta) a(-2s') z'. \\] \\noindent Cf.~ equation \\eqref{eq:def-thin} for the definition of~$Q(\\fr_1, \\ell, \\beta )$. \\par First, we note that there are $\\tilde y \\in \\Omega(y)$ an", "anchor": { "source": "manuscript", "line_start": 2873, "line_end": 2873, "char_start": 170064, "char_end": 170083 }, "origin": "source_reference", "context": "narrative" }, { "from": "lm:concentrated-surface", "to": "sec-structured-component", "type": "internal_reference", "evidence": "\\in [\\beta e^{2s'}]$, and $$\\vv w^- = X_1 \\vv v_2 + Y_1 \\vv w_2 \\in \\vv{r}^-.$$ By Lemma \\ref{lm:concentrated-surface}, we have that $\\cC = \\cC(\\cT(y), z)$ is the $\\beta^{-100} e^{-3s'}$-neighborhood of the following curve ($v_1,w_1$ as above):", "anchor": { "source": "manuscript", "line_start": 2884, "line_end": 2884, "char_start": 170588, "char_end": 170617 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:tildey-tildez", "to": "sec-structured-component", "type": "internal_reference", "evidence": "tilde X \\vv v_2 + \\tilde Y \\vv w_2 $ with $(\\tilde X, \\tilde Y) \\in \\cN^\\ell (\\vv 0)$. \\par By \\eqref{eq:tildey-tildez} and \\eqref{eq:tildez-z'}, we get \\[ \\tilde y =u(r) \\exp(\\vv w^+) \\exp(\\tilde{\\vv w}^-) u^\\ast (r^\\ast)z' \\] where $\\vv", "anchor": { "source": "manuscript", "line_start": 2907, "line_end": 2907, "char_start": 172012, "char_end": 172036 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:tildez-z'", "to": "sec-structured-component", "type": "internal_reference", "evidence": "v w_2 $ with $(\\tilde X, \\tilde Y) \\in \\cN^\\ell (\\vv 0)$. \\par By \\eqref{eq:tildey-tildez} and \\eqref{eq:tildez-z'}, we get \\[ \\tilde y =u(r) \\exp(\\vv w^+) \\exp(\\tilde{\\vv w}^-) u^\\ast (r^\\ast)z' \\] where $\\vv w^+ \\in B_{\\vv r^+}(\\beta", "anchor": { "source": "manuscript", "line_start": 2907, "line_end": 2907, "char_start": 172041, "char_end": 172061 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:pv1", "to": "sec-structured-component", "type": "internal_reference", "evidence": "$\\vv w^+ \\in B_{\\vv r^+}(\\beta e^{s'})$ with $\\|p_{\\vv v_1} (\\vv w^+)\\| \\le \\beta e^{s' - \\ell}$ by \\eqref{eq:pv1}, and $\\tilde{\\vv w}^- \\in B_{\\vv r^-}(\\beta e^{-s'})$ with $\\|p_{\\vv v_2} (\\tilde{\\vv w}^-)\\| \\le \\beta e^{-s' - \\ell}$ since $(\\", "anchor": { "source": "manuscript", "line_start": 2909, "line_end": 2909, "char_start": 172261, "char_end": 172275 }, "origin": "source_reference", "context": "narrative" }, { "from": "lm:product-w", "to": "sec-structured-component", "type": "internal_reference", "evidence": "estimates imply that \\[ \\|[\\bar{\\vv w}^-, \\bar{\\vv w}^+]\\| \\le \\beta^2 e^{-\\ell}. \\] By Lemma \\ref{lm:product-w}, we have \\[ \\exp(\\vv w^+) \\exp(\\tilde{\\vv w}^-) = \\exp(\\bar{\\vv w}) \\exp(\\bar{\\vv h}), \\] where $\\bar{\\vv w} \\in B_{\\fr_1 + \\fr_", "anchor": { "source": "manuscript", "line_start": 2913, "line_end": 2913, "char_start": 172942, "char_end": 172960 }, "origin": "source_reference", "context": "narrative" }, { "from": "lm:curve-in-N", "to": "rmk:sl2r2", "type": "internal_reference", "evidence": "l-1, \\beta ) B^{A_0 H}(\\beta) a(-2s') z'. \\] Lemma \\ref{lm:curve-in-N} roughly says that given $z \\in \\cI\\cap \\Sigma^r$ and $$ \\cN^\\ell = \\cN^\\ell(X_0, Y_0) \\subset \\fC$$ as above, all $y \\in \\fJ\\c", "anchor": { "source": "manuscript", "line_start": 2926, "line_end": 2926, "char_start": 173597, "char_end": 173616 }, "origin": "source_reference", "context": "statement" }, { "from": "prop:dimension-improvement-structured-component", "to": "sec-structured-component", "type": "internal_reference", "evidence": "ltimes \\R^2$-orbit of size $\\beta$. \\par Now we are now ready to prove Proposition \\ref{prop:dimension-improvement-structured-component} [Proof of Proposition \\ref{prop:dimension-improvement-structured-component}] \\par Assume in contradiction that \\[", "anchor": { "source": "manuscript", "line_start": 2936, "line_end": 2937, "char_start": 174019, "char_end": 174072 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:dimension-does-not-drop", "to": "prop:dimension-improvement-structured-component", "type": "candidate_proof_dependency", "evidence": "\\mu (\\bar \\cR^{\\fs}_i) > \\beta^{22}. \\] Let $\\fZ \\subset \\Pi(x)$ be the subset from Proposition \\ref{prop:dimension-does-not-drop}. By replacing $\\bar \\cR^{\\fs}_i$ with $\\bar \\cR^{\\fs}_i \\setminus \\mathfrak{Z}$, we can assume that \\eqref{eq:upper-theta", "anchor": { "source": "manuscript", "line_start": 2940, "line_end": 2940, "char_start": 174309, "char_end": 174343 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:upper-theta", "to": "prop:dimension-improvement-structured-component", "type": "candidate_proof_dependency", "evidence": "replacing $\\bar \\cR^{\\fs}_i$ with $\\bar \\cR^{\\fs}_i \\setminus \\mathfrak{Z}$, we can assume that \\eqref{eq:upper-theta} holds for any $z \\in \\cI \\cap \\bar\\cR^{\\fs}_i$. Thus we may assume: \\c", "anchor": { "source": "manuscript", "line_start": 2942, "line_end": 2942, "char_start": 174452, "char_end": 174474 }, "origin": "source_reference", "context": "proof" }, { "from": "def:structure-random-decompostion", "to": "prop:dimension-improvement-structured-component", "type": "candidate_proof_dependency", "evidence": "us fix a $z \\in \\cI\\cap \\bar \\cR^{\\fs}_i$. By the definition of $\\bar \\cR^{\\fs}_i$ (see Definition \\ref{def:structure-random-decompostion}), we have that $\\Theta'(z)$ is highly structured. Thus, there exists $r=r(z)$ such that $z \\in \\Sigma^r$ and \\", "anchor": { "source": "manuscript", "line_start": 2953, "line_end": 2953, "char_start": 174960, "char_end": 174999 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:upper-omega", "to": "prop:dimension-improvement-structured-component", "type": "candidate_proof_dependency", "evidence": "\\tilde \\mu (\\SS(z,r)) \\ge e^{-2s' - 10 C_1 \\epsilon_1 s'} Recall that by \\eqref{eq:upper-omega}, for any $y$, \\[ \\tilde \\mu(\\Omega'(y)) \\le \\beta^{-15}e^{-2 s'} e^{-2 d_i s'}. \\] Recall also that $\\SS(z,r)$ is the u", "anchor": { "source": "manuscript", "line_start": 2957, "line_end": 2957, "char_start": 175244, "char_end": 175266 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:SSzr", "to": "prop:dimension-improvement-structured-component", "type": "candidate_proof_dependency", "evidence": "z,r)$, i.e.\\ those $y\\in\\fJ\\cap\\Sigma^r$ for which $\\cC(\\cT(y), z)$ is highly concentrated. Then by \\eqref{eq:SSzr} and \\eqref{eq:upper-omega}, we have the following: \\item[$\\clubsuit$] there exist at least $e^{2d_i s' -15C_1", "anchor": { "source": "manuscript", "line_start": 2961, "line_end": 2961, "char_start": 175533, "char_end": 175548 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:upper-omega", "to": "prop:dimension-improvement-structured-component", "type": "candidate_proof_dependency", "evidence": "y\\in\\fJ\\cap\\Sigma^r$ for which $\\cC(\\cT(y), z)$ is highly concentrated. Then by \\eqref{eq:SSzr} and \\eqref{eq:upper-omega}, we have the following: \\item[$\\clubsuit$] there exist at least $e^{2d_i s' -15C_1 \\epsilon_1 s'}$ elements $", "anchor": { "source": "manuscript", "line_start": 2961, "line_end": 2961, "char_start": 175553, "char_end": 175575 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:upper-theta-2", "to": "prop:dimension-improvement-structured-component", "type": "candidate_proof_dependency", "evidence": "mph{covers}~$\\Omega'(z')$ ($z'\\in \\cI_z$) if $\\fD_z(z') \\in \\cC$. \\par To get a contradiction to \\eqref{eq:upper-theta-2}, it is enough to show that if $\\clubsuit$ holds, the union \\bigcup_{y \\in \\HS(z,r)}\\cC(\\cT", "anchor": { "source": "manuscript", "line_start": 2969, "line_end": 2969, "char_start": 176137, "char_end": 176161 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:full-omega", "to": "prop:dimension-improvement-structured-component", "type": "candidate_proof_dependency", "evidence": "(z')$ with $z' \\in\\cI_z$. Indeed, if this holds, then by the definition of full neighborhoods in~\\eqref{eq:full-omega}, we have that \\[ \\tilde \\mu(\\Theta'(z)) \\ge e^{-2s'} e^{-(d_i - 12 C_1 \\epsilon_1) s'}, \\] which contradicts \\eqref{eq:upper", "anchor": { "source": "manuscript", "line_start": 2975, "line_end": 2975, "char_start": 176561, "char_end": 176582 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:upper-theta-2", "to": "prop:dimension-improvement-structured-component", "type": "candidate_proof_dependency", "evidence": "\\[ \\tilde \\mu(\\Theta'(z)) \\ge e^{-2s'} e^{-(d_i - 12 C_1 \\epsilon_1) s'}, \\] which contradicts \\eqref{eq:upper-theta-2}. \\medskip \\par Now we aim to show that under the assumption $\\clubsuit$, \\bi", "anchor": { "source": "manuscript", "line_start": 2977, "line_end": 2977, "char_start": 176697, "char_end": 176721 }, "origin": "source_reference", "context": "proof" }, { "from": "rmk:concentrated-surface", "to": "prop:dimension-improvement-structured-component", "type": "candidate_proof_dependency", "evidence": "he proof into two cases. \\subsection*{The case \\texorpdfstring{$\\ell\\geq s'$}{l<=s'}} By Remark \\ref{rmk:concentrated-surface}.\\eqref{item:line}, in this case the highly concentrated curve neighborhoods are neighborhoods of straight lines. Without loss of", "anchor": { "source": "manuscript", "line_start": 2991, "line_end": 2991, "char_start": 177548, "char_end": 177578 }, "origin": "source_reference", "context": "proof" }, { "from": "item:line", "to": "prop:dimension-improvement-structured-component", "type": "candidate_proof_dependency", "evidence": "bsection*{The case \\texorpdfstring{$\\ell\\geq s'$}{l<=s'}} By Remark \\ref{rmk:concentrated-surface}.\\eqref{item:line}, in this case the highly concentrated curve neighborhoods are neighborhoods of straight lines. Without loss of generality, we can", "anchor": { "source": "manuscript", "line_start": 2991, "line_end": 2991, "char_start": 177579, "char_end": 177596 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:integrable-surface", "to": "prop:dimension-improvement-structured-component", "type": "candidate_proof_dependency", "evidence": "tion of highly concentrated implies that $\\|(v_1,w_1)\\|\\geq e^{s'-10C_1\\epsilon_1s'} $, hence Lemma \\ref{lm:integrable-surface} can be applied. Thus there are at most $e^{4s' + 60 C_1 \\epsilon_1 s'}$ different $y \\in \\fJ\\cap \\Sigma^r$ such that \\[ \\cC(\\c", "anchor": { "source": "manuscript", "line_start": 2992, "line_end": 2992, "char_start": 178060, "char_end": 178087 }, "origin": "source_reference", "context": "proof" }, { "from": "def:comparable", "to": "prop:dimension-improvement-structured-component", "type": "candidate_proof_dependency", "evidence": "borhoods are neighborhoods of straight lines, as well as the definition of comparable in Definition~\\ref{def:comparable}, we have that any two non-comparable curve neighborhoods $\\cC$ and $\\cC'$ passing through $(X_0, Y_0)$ do not intersect at any $(", "anchor": { "source": "manuscript", "line_start": 3017, "line_end": 3017, "char_start": 180000, "char_end": 180020 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:structure-part-theta-0", "to": "prop:dimension-improvement-structured-component", "type": "candidate_proof_dependency", "evidence": "\\par Similarly to the definition of $\\HS(z,r)$ and $\\SS(z,r)$ (cf.\\ \\eqref{eq:structure-part-theta-0} and \\eqref{eq:structure-set-theta-0}), let us define \\HS(z, r; \\cN^{s'}_1) := \\le", "anchor": { "source": "manuscript", "line_start": 3027, "line_end": 3027, "char_start": 180664, "char_end": 180697 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:structure-set-theta-0", "to": "prop:dimension-improvement-structured-component", "type": "candidate_proof_dependency", "evidence": "larly to the definition of $\\HS(z,r)$ and $\\SS(z,r)$ (cf.\\ \\eqref{eq:structure-part-theta-0} and \\eqref{eq:structure-set-theta-0}), let us define \\HS(z, r; \\cN^{s'}_1) := \\left\\{ y \\in \\fJ\\cap \\Sigma^r: \\cC(\\cT(", "anchor": { "source": "manuscript", "line_start": 3027, "line_end": 3027, "char_start": 180702, "char_end": 180734 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:curve-in-N", "to": "prop:dimension-improvement-structured-component", "type": "candidate_proof_dependency", "evidence": "(\\SS(z,r; \\cN^{s'}_{1})) \\le e^{-2s'}e^{-800 C_1 \\epsilon_1 s'}. Indeed, by Lemma \\ref{lm:curve-in-N}, there exists $z' \\in \\Theta'(z)$ so that the following holds: for any $y \\in \\HS(z, r; \\cN_1^{s'} )$, we have \\[ \\Omega'(y)", "anchor": { "source": "manuscript", "line_start": 3040, "line_end": 3040, "char_start": 181267, "char_end": 181286 }, "origin": "source_reference", "context": "proof" }, { "from": "sec-reduction", "to": "prop:dimension-improvement-structured-component", "type": "candidate_proof_dependency", "evidence": "eta ) B^{A_0 H}(\\beta) \\tilde z \\right)\\cap \\Sigma^r. \\] By our discussion at the beginning of \\S \\ref{sec-reduction}, Proposition \\ref{prop:sl2xr2-closing-lemma} holds for $\\mu_{\\cF_i}$. Thus, we have \\tilde \\mu\\left(a(2s')Q(", "anchor": { "source": "manuscript", "line_start": 3045, "line_end": 3045, "char_start": 181696, "char_end": 181715 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:sl2xr2-closing-lemma", "to": "prop:dimension-improvement-structured-component", "type": "candidate_proof_dependency", "evidence": "right)\\cap \\Sigma^r. \\] By our discussion at the beginning of \\S \\ref{sec-reduction}, Proposition \\ref{prop:sl2xr2-closing-lemma} holds for $\\mu_{\\cF_i}$. Thus, we have \\tilde \\mu\\left(a(2s')Q(\\fr_1, s'-1, \\beta ) B^{A_0 H}(\\beta) \\tilde", "anchor": { "source": "manuscript", "line_start": 3045, "line_end": 3045, "char_start": 181729, "char_end": 181760 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:sszrN-small", "to": "prop:dimension-improvement-structured-component", "type": "candidate_proof_dependency", "evidence": "H}(\\beta) \\tilde z\\right) \\\\ &\\le e^{-2s'}e^{-800 C_1 \\epsilon_1 s'}. This proves \\eqref{eq:sszrN-small}. Thus, $\\clubsuit$ holds with $\\HS(z,r)$ replaced by $$\\HS(z,r) \\setminus \\HS(z, r;\\cN^{s'}_{1}).$$ By repeating the above", "anchor": { "source": "manuscript", "line_start": 3056, "line_end": 3056, "char_start": 182655, "char_end": 182677 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:nowweaim", "to": "prop:dimension-improvement-structured-component", "type": "candidate_proof_dependency", "evidence": "n_1 s'} \\ge e^{d_i s' + 20 C_1 \\epsilon_1 s'}$$ different full $\\Omega'(z')$'s. This establishes \\eqref{eq:nowweaim}, completing the proof for the case $\\ell \\ge s'$. \\subsection*{The case \\texorpdfstring{$\\ell < s'$}{ls'$ case, and to deal with this case we shall need to use Proposition~\\ref{prop:sl2xr2-closing-lemma} (which roughly says that our Diophantine assumptions guarantee that $\\tilde\\mu$ does not concentrate along orbits of $\\SL(2,R)\\lt", "anchor": { "source": "manuscript", "line_start": 3181, "line_end": 3181, "char_start": 191996, "char_end": 192027 }, "origin": "source_reference", "context": "proof" }, { "from": "lm:curve-in-N", "to": "prop:dimension-improvement-structured-component", "type": "candidate_proof_dependency", "evidence": "In fact, by Lemma \\ref{lm:curve-in-N}, there exists $z' \\in \\Theta'(z)$ so that the following holds: For any $y \\in \\fJ\\cap \\Sigma^r$ such that $\\cC(\\cT(y), z) \\subse", "anchor": { "source": "manuscript", "line_start": 3197, "line_end": 3197, "char_start": 193256, "char_end": 193275 }, "origin": "source_reference", "context": "proof" }, { "from": "sec-reduction", "to": "prop:dimension-improvement-structured-component", "type": "candidate_proof_dependency", "evidence": "l-1, \\beta ) B^{A_0 H}(\\beta) \\tilde z \\cap \\Sigma^r. \\] By our discussion at the beginning of \\S \\ref{sec-reduction}, Proposition \\ref{prop:sl2xr2-closing-lemma} holds for $\\mu_{\\cF_i}$. Thus, we have \\tilde \\mu \\left(a(2s')Q", "anchor": { "source": "manuscript", "line_start": 3203, "line_end": 3203, "char_start": 193739, "char_end": 193758 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:sl2xr2-closing-lemma", "to": "prop:dimension-improvement-structured-component", "type": "candidate_proof_dependency", "evidence": "lde z \\cap \\Sigma^r. \\] By our discussion at the beginning of \\S \\ref{sec-reduction}, Proposition \\ref{prop:sl2xr2-closing-lemma} holds for $\\mu_{\\cF_i}$. Thus, we have \\tilde \\mu \\left(a(2s')Q(\\fr_1, \\ell-1, \\beta ) B^{A_0 H}(\\beta) \\til", "anchor": { "source": "manuscript", "line_start": 3203, "line_end": 3203, "char_start": 193772, "char_end": 193803 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:high-dimension-to-equidistribution-a", "to": "sec-main-proof", "type": "internal_reference", "evidence": "\\section{Proof of the main theorem} \\par In order to apply Proposition \\ref{prop:high-dimension-to-equidistribution-a}, we first prove the following statement: \\par Let $s \\le \\hat s < t$, $s' =(", "anchor": { "source": "manuscript", "line_start": 3242, "line_end": 3242, "char_start": 196176, "char_end": 196223 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:high-dimension-to-equidistribution-a", "to": "prop:high-dim-decomposition", "type": "internal_reference", "evidence": "ity measure on $[\\beta]^2$ with dimension larger than $2-2\\theta$ at scale $s'$ (see Proposition \\ref{prop:high-dimension-to-equidistribution-a} for the definition). \\par Let us define \\Upsil", "anchor": { "source": "manuscript", "line_start": 3258, "line_end": 3258, "char_start": 197324, "char_end": 197371 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:fvy", "to": "prop:high-dim-decomposition", "type": "candidate_proof_dependency", "evidence": "{eq:bary-y-Qs} \\bar{y} \\in Q_{s'}(\\beta) B^{A_0 H}(\\beta)y. Moreover, we may apply \\eqref{eq:fvy}, \\eqref{eq:fwy}, and \\eqref{eq:fy} to conclude that for every $r' \\in [\\beta]$, there exists $f(r')$ such that $|r' - f(r')| \\le", "anchor": { "source": "manuscript", "line_start": 3305, "line_end": 3305, "char_start": 199901, "char_end": 199915 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:fwy", "to": "prop:high-dim-decomposition", "type": "candidate_proof_dependency", "evidence": "\\bar{y} \\in Q_{s'}(\\beta) B^{A_0 H}(\\beta)y. Moreover, we may apply \\eqref{eq:fvy}, \\eqref{eq:fwy}, and \\eqref{eq:fy} to conclude that for every $r' \\in [\\beta]$, there exists $f(r')$ such that $|r' - f(r')| \\le \\beta|r'|$, and", "anchor": { "source": "manuscript", "line_start": 3305, "line_end": 3305, "char_start": 199917, "char_end": 199931 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:fy", "to": "prop:high-dim-decomposition", "type": "candidate_proof_dependency", "evidence": "\\beta) B^{A_0 H}(\\beta)y. Moreover, we may apply \\eqref{eq:fvy}, \\eqref{eq:fwy}, and \\eqref{eq:fy} to conclude that for every $r' \\in [\\beta]$, there exists $f(r')$ such that $|r' - f(r')| \\le \\beta|r'|$, and \\[ a(2s')u\\left(f(", "anchor": { "source": "manuscript", "line_start": 3305, "line_end": 3305, "char_start": 199937, "char_end": 199950 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:compare-u-orbit", "to": "prop:high-dim-decomposition", "type": "candidate_proof_dependency", "evidence": "normalized Lebesgue measure on the union $$ \\bigcup_{\\bar y \\in \\mathcal T}u([\\beta])\\bar y.$$ By \\eqref{eq:compare-u-orbit}, we have that $a(2s')_\\ast\\mu_k$ and $a(2s')_{\\ast}\\bar \\mu_k$ are $O(\\beta)$-close, namely, for any $f \\in C_c^{\\infty} (X)$, \\[", "anchor": { "source": "manuscript", "line_start": 3315, "line_end": 3315, "char_start": 200568, "char_end": 200594 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:bary-y-Qs", "to": "prop:high-dim-decomposition", "type": "candidate_proof_dependency", "evidence": "\\dd r. \\] We claim that $\\rho_k$ has dimension larger than $2-2\\theta$ at scale $s'$. In fact, by \\eqref{eq:bary-y-Qs}, we have that for $y$ and $\\bar y$ constructed as above, $$u([\\beta])\\bar y \\in Q_{s'}(\\beta) B^{A_0 H}(\\beta)y.$$ Therefore, giv", "anchor": { "source": "manuscript", "line_start": 3321, "line_end": 3321, "char_start": 201229, "char_end": 201249 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:first-decompose-hat-mu", "to": "prop:high-dim-decomposition", "type": "candidate_proof_dependency", "evidence": "where $(v_S, w_S)$ is the center of $S$. This proves the claim. \\par Combining the claim above with \\eqref{eq:first-decompose-hat-mu} and \\eqref{eq:compare-u-orbit}, we conclude \\eqref{eq:decompose-hat-mu}.", "anchor": { "source": "manuscript", "line_start": 3326, "line_end": 3326, "char_start": 201747, "char_end": 201780 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:compare-u-orbit", "to": "prop:high-dim-decomposition", "type": "candidate_proof_dependency", "evidence": "$. This proves the claim. \\par Combining the claim above with \\eqref{eq:first-decompose-hat-mu} and \\eqref{eq:compare-u-orbit}, we conclude \\eqref{eq:decompose-hat-mu}.", "anchor": { "source": "manuscript", "line_start": 3326, "line_end": 3326, "char_start": 201785, "char_end": 201811 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:decompose-hat-mu", "to": "prop:high-dim-decomposition", "type": "candidate_proof_dependency", "evidence": "the claim above with \\eqref{eq:first-decompose-hat-mu} and \\eqref{eq:compare-u-orbit}, we conclude \\eqref{eq:decompose-hat-mu}.", "anchor": { "source": "manuscript", "line_start": 3326, "line_end": 3326, "char_start": 201825, "char_end": 201852 }, "origin": "source_reference", "context": "proof" }, { "from": "thm:main-thm", "to": "sec-main-proof", "type": "internal_reference", "evidence": "\\par We are now equipped to prove Theorem \\ref{thm:main-thm}. [Proof of Theorem \\ref{thm:main-thm}] \\par By Proposition \\ref{prop:initial-bound-dimension}, we have that $\\mu_{\\", "anchor": { "source": "manuscript", "line_start": 3340, "line_end": 3340, "char_start": 202798, "char_end": 202816 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:initial-bound-dimension", "to": "thm:main-thm", "type": "candidate_proof_dependency", "evidence": "Theorem \\ref{thm:main-thm}. [Proof of Theorem \\ref{thm:main-thm}] \\par By Proposition \\ref{prop:initial-bound-dimension}, we have that $\\mu_{\\cF_0}$ is $(d_0, s_0)$-good. By Proposition \\ref{prop:dimension-improvement-random}, \\ref{prop:dimension-imp", "anchor": { "source": "manuscript", "line_start": 3342, "line_end": 3342, "char_start": 202890, "char_end": 202924 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:dimension-improvement-random", "to": "thm:main-thm", "type": "candidate_proof_dependency", "evidence": "\\ref{prop:initial-bound-dimension}, we have that $\\mu_{\\cF_0}$ is $(d_0, s_0)$-good. By Proposition \\ref{prop:dimension-improvement-random}, \\ref{prop:dimension-improvement-strange-part}, and \\ref{prop:dimension-improvement-structured-component} we get Proposition \\ref", "anchor": { "source": "manuscript", "line_start": 3342, "line_end": 3342, "char_start": 202990, "char_end": 203029 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:dimension-improvement-strange-part", "to": "thm:main-thm", "type": "candidate_proof_dependency", "evidence": "ve that $\\mu_{\\cF_0}$ is $(d_0, s_0)$-good. By Proposition \\ref{prop:dimension-improvement-random}, \\ref{prop:dimension-improvement-strange-part}, and \\ref{prop:dimension-improvement-structured-component} we get Proposition \\ref{prop:improve-dimension}. By Proposition \\ref{p", "anchor": { "source": "manuscript", "line_start": 3342, "line_end": 3342, "char_start": 203031, "char_end": 203076 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:dimension-improvement-structured-component", "to": "thm:main-thm", "type": "candidate_proof_dependency", "evidence": "osition \\ref{prop:dimension-improvement-random}, \\ref{prop:dimension-improvement-strange-part}, and \\ref{prop:dimension-improvement-structured-component} we get Proposition \\ref{prop:improve-dimension}. By Proposition \\ref{prop:reduction}, it implies that $\\mu_{\\cF_{i}}$ is $(d_i, s", "anchor": { "source": "manuscript", "line_start": 3342, "line_end": 3342, "char_start": 203082, "char_end": 203135 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:improve-dimension", "to": "thm:main-thm", "type": "candidate_proof_dependency", "evidence": "ovement-strange-part}, and \\ref{prop:dimension-improvement-structured-component} we get Proposition \\ref{prop:improve-dimension}. By Proposition \\ref{prop:reduction}, it implies that $\\mu_{\\cF_{i}}$ is $(d_i, s_i)$-good, where $d_i = d_0 + i \\epsilon_2$ and", "anchor": { "source": "manuscript", "line_start": 3342, "line_end": 3342, "char_start": 203155, "char_end": 203183 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:reduction", "to": "thm:main-thm", "type": "candidate_proof_dependency", "evidence": "n-improvement-structured-component} we get Proposition \\ref{prop:improve-dimension}. By Proposition \\ref{prop:reduction}, it implies that $\\mu_{\\cF_{i}}$ is $(d_i, s_i)$-good, where $d_i = d_0 + i \\epsilon_2$ and $s_i = 2^{-i}s_0$(we remove an expone", "anchor": { "source": "manuscript", "line_start": 3342, "line_end": 3342, "char_start": 203200, "char_end": 203220 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:high-dim-decomposition", "to": "thm:main-thm", "type": "candidate_proof_dependency", "evidence": "step). We can proceed with this inductive construction until $d_i \\ge 4 - \\theta$. By Proposition \\ref{prop:high-dim-decomposition}, we have that \\[ a(2s_i)_\\ast \\mu_{\\cF_i} = \\sum_{k} \\nu_k a(2s_i)_\\ast \\mu_k + O(\\beta).\\] Note that by Proposition \\ref{prop:", "anchor": { "source": "manuscript", "line_start": 3343, "line_end": 3343, "char_start": 203483, "char_end": 203516 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:high-dim-decomposition", "to": "thm:main-thm", "type": "candidate_proof_dependency", "evidence": "a(2s_i)_\\ast \\mu_{\\cF_i} = \\sum_{k} \\nu_k a(2s_i)_\\ast \\mu_k + O(\\beta).\\] Note that by Proposition \\ref{prop:high-dim-decomposition}, every $\\mu_k$ is defined by a measure $\\rho_k$ on $[\\beta]^2$ with dimension larger than $2-2\\theta$ at scale $s'$. Then Propos", "anchor": { "source": "manuscript", "line_start": 3345, "line_end": 3345, "char_start": 203636, "char_end": 203669 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:high-dimension-to-equidistribution-a", "to": "thm:main-thm", "type": "candidate_proof_dependency", "evidence": "sure $\\rho_k$ on $[\\beta]^2$ with dimension larger than $2-2\\theta$ at scale $s'$. Then Proposition \\ref{prop:high-dimension-to-equidistribution-a}, we have that $a(2 s_i)_\\ast \\mu_{k}$ is effectively equidistributed. This implies that $a(2s_i)_\\ast \\mu_{\\cF_i}$ is effective", "anchor": { "source": "manuscript", "line_start": 3347, "line_end": 3347, "char_start": 203805, "char_end": 203852 }, "origin": "source_reference", "context": "proof" }, { "from": "external:ratner_2", "to": "sec-intro", "type": "external_reference", "evidence": "\\section{Introduction} \\subsection{Ratner's theorem and its effective versions} \\par In 1991, Ratner \\cite{ratner_2} proved the following fundamental theorem on equidistribution of unipotent orbits in homogeneous spaces: Let $G$ be a Lie group and $\\Gamma$ be a lattice in $G$, na", "anchor": { "source": "manuscript", "line_start": 381, "line_end": 381, "char_start": 10063, "char_end": 10078 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:ratner-acta", "to": "sec-intro", "type": "external_reference", "evidence": "(u(r)x) \\dd r = \\int_{Lx} f \\dd \\mu_L .\\] \\par Since Ratner's proof relies on her measure classification theorem (cf. \\cite{ratner-acta}, \\cite{ratner-invent}, \\cite{Ratner}) which proves a conjecture by Dani \\cite{Dani1981}, it does not tell how fast it tends to equidistribution. Therefore, a natural question is whether we can make", "anchor": { "source": "manuscript", "line_start": 394, "line_end": 394, "char_start": 11007, "char_end": 11025 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:ratner-invent", "to": "sec-intro", "type": "external_reference", "evidence": "t_{Lx} f \\dd \\mu_L .\\] \\par Since Ratner's proof relies on her measure classification theorem (cf. \\cite{ratner-acta}, \\cite{ratner-invent}, \\cite{Ratner}) which proves a conjecture by Dani \\cite{Dani1981}, it does not tell how fast it tends to equidistribution. Therefore, a natural question is whether we can make the equidistribution", "anchor": { "source": "manuscript", "line_start": 394, "line_end": 394, "char_start": 11027, "char_end": 11047 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:Ratner", "to": "sec-intro", "type": "external_reference", "evidence": "\\par Since Ratner's proof relies on her measure classification theorem (cf. \\cite{ratner-acta}, \\cite{ratner-invent}, \\cite{Ratner}) which proves a conjecture by Dani \\cite{Dani1981}, it does not tell how fast it tends to equidistribution. Therefore, a natural question is whether we can make the equidistribution effective, name", "anchor": { "source": "manuscript", "line_start": 394, "line_end": 394, "char_start": 11049, "char_end": 11062 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:Dani1981", "to": "sec-intro", "type": "external_reference", "evidence": "r measure classification theorem (cf. \\cite{ratner-acta}, \\cite{ratner-invent}, \\cite{Ratner}) which proves a conjecture by Dani \\cite{Dani1981}, it does not tell how fast it tends to equidistribution. Therefore, a natural question is whether we can make the equidistribution effective, namely, giving an explicit upper bound on the differenc", "anchor": { "source": "manuscript", "line_start": 394, "line_end": 394, "char_start": 11098, "char_end": 11113 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:lindenstrauss-margulis", "to": "sec-intro", "type": "external_reference", "evidence": "for given $T >0$. Moreover, effective versions of equidistribution of $U$-orbits have many applications to number theory, see \\cite{lindenstrauss-margulis}, \\cite{einsiedler-mohammadi}, \\cite{LM-preprint2021}, \\cite{LMW-preprint2022}, \\cite{Venkatesh2010}, \\cite{nelson-venkatesh}, \\cite{einsiedler-margulis-venkatesh}, \\cite{chow-yang}, \\cite{Browning2", "anchor": { "source": "manuscript", "line_start": 400, "line_end": 400, "char_start": 11537, "char_end": 11566 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:einsiedler-mohammadi", "to": "sec-intro", "type": "external_reference", "evidence": "fective versions of equidistribution of $U$-orbits have many applications to number theory, see \\cite{lindenstrauss-margulis}, \\cite{einsiedler-mohammadi}, \\cite{LM-preprint2021}, \\cite{LMW-preprint2022}, \\cite{Venkatesh2010}, \\cite{nelson-venkatesh}, \\cite{einsiedler-margulis-venkatesh}, \\cite{chow-yang}, \\cite{Browning2016} and references therein", "anchor": { "source": "manuscript", "line_start": 400, "line_end": 400, "char_start": 11568, "char_end": 11595 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:LM-preprint2021", "to": "sec-intro", "type": "external_reference", "evidence": "ibution of $U$-orbits have many applications to number theory, see \\cite{lindenstrauss-margulis}, \\cite{einsiedler-mohammadi}, \\cite{LM-preprint2021}, \\cite{LMW-preprint2022}, \\cite{Venkatesh2010}, \\cite{nelson-venkatesh}, \\cite{einsiedler-margulis-venkatesh}, \\cite{chow-yang}, \\cite{Browning2016} and references therein for details. As a resul", "anchor": { "source": "manuscript", "line_start": 400, "line_end": 400, "char_start": 11597, "char_end": 11619 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:LMW-preprint2022", "to": "sec-intro", "type": "external_reference", "evidence": "ve many applications to number theory, see \\cite{lindenstrauss-margulis}, \\cite{einsiedler-mohammadi}, \\cite{LM-preprint2021}, \\cite{LMW-preprint2022}, \\cite{Venkatesh2010}, \\cite{nelson-venkatesh}, \\cite{einsiedler-margulis-venkatesh}, \\cite{chow-yang}, \\cite{Browning2016} and references therein for details. As a result, proving effective vers", "anchor": { "source": "manuscript", "line_start": 400, "line_end": 400, "char_start": 11621, "char_end": 11644 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:Venkatesh2010", "to": "sec-intro", "type": "external_reference", "evidence": "ber theory, see \\cite{lindenstrauss-margulis}, \\cite{einsiedler-mohammadi}, \\cite{LM-preprint2021}, \\cite{LMW-preprint2022}, \\cite{Venkatesh2010}, \\cite{nelson-venkatesh}, \\cite{einsiedler-margulis-venkatesh}, \\cite{chow-yang}, \\cite{Browning2016} and references therein for details. As a result, proving effective versions of Ratner's theorem", "anchor": { "source": "manuscript", "line_start": 401, "line_end": 401, "char_start": 11648, "char_end": 11668 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:nelson-venkatesh", "to": "sec-intro", "type": "external_reference", "evidence": "te{lindenstrauss-margulis}, \\cite{einsiedler-mohammadi}, \\cite{LM-preprint2021}, \\cite{LMW-preprint2022}, \\cite{Venkatesh2010}, \\cite{nelson-venkatesh}, \\cite{einsiedler-margulis-venkatesh}, \\cite{chow-yang}, \\cite{Browning2016} and references therein for details. As a result, proving effective versions of Ratner's theorem has attracted much att", "anchor": { "source": "manuscript", "line_start": 401, "line_end": 401, "char_start": 11670, "char_end": 11693 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:einsiedler-margulis-venkatesh", "to": "sec-intro", "type": "external_reference", "evidence": "}, \\cite{einsiedler-mohammadi}, \\cite{LM-preprint2021}, \\cite{LMW-preprint2022}, \\cite{Venkatesh2010}, \\cite{nelson-venkatesh}, \\cite{einsiedler-margulis-venkatesh}, \\cite{chow-yang}, \\cite{Browning2016} and references therein for details. As a result, proving effective versions of Ratner's theorem has attracted much attention and has been a major challenge", "anchor": { "source": "manuscript", "line_start": 401, "line_end": 401, "char_start": 11695, "char_end": 11731 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:chow-yang", "to": "sec-intro", "type": "external_reference", "evidence": "LM-preprint2021}, \\cite{LMW-preprint2022}, \\cite{Venkatesh2010}, \\cite{nelson-venkatesh}, \\cite{einsiedler-margulis-venkatesh}, \\cite{chow-yang}, \\cite{Browning2016} and references therein for details. As a result, proving effective versions of Ratner's theorem has attracted much attention and has been a major challenge in homogeneous dyn", "anchor": { "source": "manuscript", "line_start": 401, "line_end": 401, "char_start": 11733, "char_end": 11749 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:Browning2016", "to": "sec-intro", "type": "external_reference", "evidence": "\\cite{LMW-preprint2022}, \\cite{Venkatesh2010}, \\cite{nelson-venkatesh}, \\cite{einsiedler-margulis-venkatesh}, \\cite{chow-yang}, \\cite{Browning2016} and references therein for details. As a result, proving effective versions of Ratner's theorem has attracted much attention and has been a major challenge in homogeneous dynamics. \\par In rece", "anchor": { "source": "manuscript", "line_start": 401, "line_end": 401, "char_start": 11751, "char_end": 11770 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:green_tao2012", "to": "sec-intro", "type": "external_reference", "evidence": "ive equidistribution results. For $G$ being nilpotent, the effective version of Ratner's theorem was established by Green and Tao \\cite{green_tao2012}. For $G = \\SL(2, \\R)$, since the one-parameter unipotent subgroup is horospherical with respect to the diagonal subgroup, one can apply the thickening argument developed in Margulis's thesis (cf.", "anchor": { "source": "manuscript", "line_start": 405, "line_end": 405, "char_start": 12172, "char_end": 12192 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:margulis-thesis", "to": "sec-intro", "type": "external_reference", "evidence": "horospherical with respect to the diagonal subgroup, one can apply the thickening argument developed in Margulis's thesis (cf. \\cite{margulis-thesis}, \\cite{Klein-Mar-Effective-Equid}) and spectral gap results for unitary representations of semisimple groups to establish effective equidistribution. See \\cite{sarnak1982}, \\cite{burger1990}, \\", "anchor": { "source": "manuscript", "line_start": 407, "line_end": 407, "char_start": 12393, "char_end": 12415 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:Klein-Mar-Effective-Equid", "to": "sec-intro", "type": "external_reference", "evidence": "ect to the diagonal subgroup, one can apply the thickening argument developed in Margulis's thesis (cf. \\cite{margulis-thesis}, \\cite{Klein-Mar-Effective-Equid}) and spectral gap results for unitary representations of semisimple groups to establish effective equidistribution. See \\cite{sarnak1982}, \\cite{burger1990}, \\cite{Flaminio-Forni}, \\cite{Sarnak", "anchor": { "source": "manuscript", "line_start": 407, "line_end": 407, "char_start": 12417, "char_end": 12449 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:sarnak1982", "to": "sec-intro", "type": "external_reference", "evidence": "id}) and spectral gap results for unitary representations of semisimple groups to establish effective equidistribution. See \\cite{sarnak1982}, \\cite{burger1990}, \\cite{Flaminio-Forni}, \\cite{Sarnak-Ubis}, \\cite{Strombergsson2004} and \\cite{Strombergsson2013} for works in this setting. In fact, equidistribution for this case was proved be", "anchor": { "source": "manuscript", "line_start": 410, "line_end": 410, "char_start": 12576, "char_end": 12593 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:burger1990", "to": "sec-intro", "type": "external_reference", "evidence": "gap results for unitary representations of semisimple groups to establish effective equidistribution. See \\cite{sarnak1982}, \\cite{burger1990}, \\cite{Flaminio-Forni}, \\cite{Sarnak-Ubis}, \\cite{Strombergsson2004} and \\cite{Strombergsson2013} for works in this setting. In fact, equidistribution for this case was proved before Ratner's theor", "anchor": { "source": "manuscript", "line_start": 410, "line_end": 410, "char_start": 12595, "char_end": 12612 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:Flaminio-Forni", "to": "sec-intro", "type": "external_reference", "evidence": "itary representations of semisimple groups to establish effective equidistribution. See \\cite{sarnak1982}, \\cite{burger1990}, \\cite{Flaminio-Forni}, \\cite{Sarnak-Ubis}, \\cite{Strombergsson2004} and \\cite{Strombergsson2013} for works in this setting. In fact, equidistribution for this case was proved before Ratner's theorem, see \\cite{furstenbe", "anchor": { "source": "manuscript", "line_start": 410, "line_end": 410, "char_start": 12614, "char_end": 12635 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:Sarnak-Ubis", "to": "sec-intro", "type": "external_reference", "evidence": "f semisimple groups to establish effective equidistribution. See \\cite{sarnak1982}, \\cite{burger1990}, \\cite{Flaminio-Forni}, \\cite{Sarnak-Ubis}, \\cite{Strombergsson2004} and \\cite{Strombergsson2013} for works in this setting. In fact, equidistribution for this case was proved before Ratner's theorem, see \\cite{furstenberg1973}, \\cite{dani-", "anchor": { "source": "manuscript", "line_start": 410, "line_end": 410, "char_start": 12637, "char_end": 12655 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:Strombergsson2004", "to": "sec-intro", "type": "external_reference", "evidence": "to establish effective equidistribution. See \\cite{sarnak1982}, \\cite{burger1990}, \\cite{Flaminio-Forni}, \\cite{Sarnak-Ubis}, \\cite{Strombergsson2004} and \\cite{Strombergsson2013} for works in this setting. In fact, equidistribution for this case was proved before Ratner's theorem, see \\cite{furstenberg1973}, \\cite{dani-smillie}. Using similar", "anchor": { "source": "manuscript", "line_start": 410, "line_end": 410, "char_start": 12657, "char_end": 12681 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:Strombergsson2013", "to": "sec-intro", "type": "external_reference", "evidence": "distribution. See \\cite{sarnak1982}, \\cite{burger1990}, \\cite{Flaminio-Forni}, \\cite{Sarnak-Ubis}, \\cite{Strombergsson2004} and \\cite{Strombergsson2013} for works in this setting. In fact, equidistribution for this case was proved before Ratner's theorem, see \\cite{furstenberg1973}, \\cite{dani-smillie}. Using similar arguments one can also establ", "anchor": { "source": "manuscript", "line_start": 410, "line_end": 410, "char_start": 12686, "char_end": 12710 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:furstenberg1973", "to": "sec-intro", "type": "external_reference", "evidence": "e{Strombergsson2013} for works in this setting. In fact, equidistribution for this case was proved before Ratner's theorem, see \\cite{furstenberg1973}, \\cite{dani-smillie}. Using similar arguments one can also establish effective equidistribution results for horospherical unipotent orbits in homogeneous spaces. See \\cite{Klein-Mar-Effective-Equ", "anchor": { "source": "manuscript", "line_start": 411, "line_end": 411, "char_start": 12820, "char_end": 12842 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:dani-smillie", "to": "sec-intro", "type": "external_reference", "evidence": "works in this setting. In fact, equidistribution for this case was proved before Ratner's theorem, see \\cite{furstenberg1973}, \\cite{dani-smillie}. Using similar arguments one can also establish effective equidistribution results for horospherical unipotent orbits in homogeneous spaces. See \\cite{Klein-Mar-Effective-Equid}, \\cite{edwards-ef", "anchor": { "source": "manuscript", "line_start": 411, "line_end": 411, "char_start": 12844, "char_end": 12863 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:Klein-Mar-Effective-Equid", "to": "sec-intro", "type": "external_reference", "evidence": "guments one can also establish effective equidistribution results for horospherical unipotent orbits in homogeneous spaces. See \\cite{Klein-Mar-Effective-Equid}, \\cite{edwards-effective} for results in this setting. Using techniques from analytic number theory, Strombergsson \\cite{Strombergsson2015} established the effective equidistribution for $G/\\Gamm", "anchor": { "source": "manuscript", "line_start": 413, "line_end": 413, "char_start": 13013, "char_end": 13045 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:edwards-effective", "to": "sec-intro", "type": "external_reference", "evidence": "ective equidistribution results for horospherical unipotent orbits in homogeneous spaces. See \\cite{Klein-Mar-Effective-Equid}, \\cite{edwards-effective} for results in this setting. Using techniques from analytic number theory, Strombergsson \\cite{Strombergsson2015} established the effective equidistribution for $G/\\Gamma = \\SL(2, \\R) \\ltimes \\R^", "anchor": { "source": "manuscript", "line_start": 413, "line_end": 413, "char_start": 13047, "char_end": 13071 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:Strombergsson2015", "to": "sec-intro", "type": "external_reference", "evidence": "ctive-Equid}, \\cite{edwards-effective} for results in this setting. Using techniques from analytic number theory, Strombergsson \\cite{Strombergsson2015} established the effective equidistribution for $G/\\Gamma = \\SL(2, \\R) \\ltimes \\R^2/ \\SL(2,\\Z) \\ltimes \\Z^2$ and $U$ being a one-parameter unipotent (and horospherical) subgroup in the semisimple", "anchor": { "source": "manuscript", "line_start": 414, "line_end": 414, "char_start": 13163, "char_end": 13187 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:chow-yang", "to": "sec-intro", "type": "external_reference", "evidence": "-parameter unipotent (and horospherical) subgroup in the semisimple part. Building on Strombergsson's result, Chow and the author \\cite{chow-yang} proved an effective equidistribution result for a special family of one-parameter unipotent orbits in $G/\\Gamma = \\SL(3,\\R)/\\SL(3,\\Z)$. As an application, we proved that Gallagher's theorem in", "anchor": { "source": "manuscript", "line_start": 416, "line_end": 416, "char_start": 13449, "char_end": 13465 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:Browning2016", "to": "sec-intro", "type": "external_reference", "evidence": "cative Diophantine approximation holds for almost every point on any given planar straight line. The reader is also referred to \\cite{Browning2016} for a similar effective equidistibution result which has applications to number theory. Strombergsson's result was recently generalized by Kim \\cite{kim2021} to $\\SL(n, \\R) \\ltimes \\R^n/ \\SL(n,\\Z) \\", "anchor": { "source": "manuscript", "line_start": 420, "line_end": 420, "char_start": 13803, "char_end": 13822 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:kim2021", "to": "sec-intro", "type": "external_reference", "evidence": "effective equidistibution result which has applications to number theory. Strombergsson's result was recently generalized by Kim \\cite{kim2021} to $\\SL(n, \\R) \\ltimes \\R^n/ \\SL(n,\\Z) \\ltimes \\Z^n$ with the unipotent subgroup being horospherical in the semisimple part. Recently, Lindenstrauss, Mohammadi and Wang \\cite{LMW-preprint2022} es", "anchor": { "source": "manuscript", "line_start": 421, "line_end": 421, "char_start": 13967, "char_end": 13981 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:LMW-preprint2022", "to": "sec-intro", "type": "external_reference", "evidence": "mes \\Z^n$ with the unipotent subgroup being horospherical in the semisimple part. Recently, Lindenstrauss, Mohammadi and Wang \\cite{LMW-preprint2022} established effective Ratner's theorem for one-parameter unipotent orbits in $G/\\Gamma$ where $G = \\SL(2, \\R) \\times \\SL(2, \\R)$ or $\\SL(2,\\C)$. The reader is referred to \\cite{einsiedler-marguli", "anchor": { "source": "manuscript", "line_start": 423, "line_end": 423, "char_start": 14155, "char_end": 14178 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:einsiedler-margulis-venkatesh", "to": "sec-intro", "type": "external_reference", "evidence": "one-parameter unipotent orbits in $G/\\Gamma$ where $G = \\SL(2, \\R) \\times \\SL(2, \\R)$ or $\\SL(2,\\C)$. The reader is referred to \\cite{einsiedler-margulis-venkatesh} and \\cite{einsiedler-margulis-mohammadi-venkatesh} for effective equidistribution of closed orbits of maximal semisimple subgroups which can be regarded as an effective version of a result by Moze", "anchor": { "source": "manuscript", "line_start": 425, "line_end": 425, "char_start": 14354, "char_end": 14390 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:einsiedler-margulis-mohammadi-venkatesh", "to": "sec-intro", "type": "external_reference", "evidence": "ma$ where $G = \\SL(2, \\R) \\times \\SL(2, \\R)$ or $\\SL(2,\\C)$. The reader is referred to \\cite{einsiedler-margulis-venkatesh} and \\cite{einsiedler-margulis-mohammadi-venkatesh} for effective equidistribution of closed orbits of maximal semisimple subgroups which can be regarded as an effective version of a result by Mozes and Shah \\cite{Mozes_Shah}. \\par Concerning effe", "anchor": { "source": "manuscript", "line_start": 425, "line_end": 426, "char_start": 14395, "char_end": 14441 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:Mozes_Shah", "to": "sec-intro", "type": "external_reference", "evidence": "on of closed orbits of maximal semisimple subgroups which can be regarded as an effective version of a result by Mozes and Shah \\cite{Mozes_Shah}. \\par Concerning effective density, there are also significant results. Lindenstrauss and Margulis proved \\cite{lindenstrauss-margulis} an effective density result for one-parameter unipotent orbi", "anchor": { "source": "manuscript", "line_start": 427, "line_end": 427, "char_start": 14601, "char_end": 14618 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:lindenstrauss-margulis", "to": "sec-intro", "type": "external_reference", "evidence": "Shah \\cite{Mozes_Shah}. \\par Concerning effective density, there are also significant results. Lindenstrauss and Margulis proved \\cite{lindenstrauss-margulis} an effective density result for one-parameter unipotent orbits in $\\SL(3,\\R)/\\SL(3,\\Z)$ and applied it to prove an effective version of Oppenheim's conjecture. Recently, Lindenstrauss and Mohammadi", "anchor": { "source": "manuscript", "line_start": 428, "line_end": 428, "char_start": 14726, "char_end": 14755 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:LM-preprint2021", "to": "sec-intro", "type": "external_reference", "evidence": "SL(3,\\R)/\\SL(3,\\Z)$ and applied it to prove an effective version of Oppenheim's conjecture. Recently, Lindenstrauss and Mohammadi \\cite{LM-preprint2021} established an optimal effective density result or unipotent orbits in $G/\\Gamma$ for $G = \\SL(2, \\R) \\times \\SL(2, \\R)$ or $\\SL(2, \\C)$. \\subsection{Notation} \\par Through", "anchor": { "source": "manuscript", "line_start": 430, "line_end": 430, "char_start": 14956, "char_end": 14978 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:KNSY", "to": "rmk:expanding-line", "type": "external_reference", "evidence": "{proof} The qualitative version of Corollary \\ref{cor:straight-line} was proved in \\cite{KNSY} for $\\omega(\\vv w) \\le 2$ using Ratner's theorem. The authors also proved that the equidistribution does not hold for $\\omega(\\vv w) > 2$.", "anchor": { "source": "manuscript", "line_start": 550, "line_end": 550, "char_start": 20887, "char_end": 20898 }, "origin": "source_reference", "locator": "", "context": "statement" }, { "from": "external:Bernik-Kleinbock-Margulis", "to": "thm:main-thm", "type": "external_reference", "evidence": "ll} := \\left\\{ r_0\\in [1]:a_0(\\ell)a_0(t/2) a_1(t/2) z(-\\psi_2'(r_0)) n(\\psi(r_0)) \\Gamma \\not\\in X_{e^{-t/10}} \\right\\}.\\] By \\cite[Theorem 1.4]{Bernik-Kleinbock-Margulis}, we have that $$\\Leb\\left(\\fm_{\\ell}\\right) = O(e^{-\\eta_2 t})$$ for some constant $\\eta_2 >0$ independent of $\\ell$. Let us remove all $\\fm_{\\ell}$ from $[1]$ and get a subset $\\fm \\subset", "anchor": { "source": "manuscript", "line_start": 587, "line_end": 587, "char_start": 23058, "char_end": 23103 }, "origin": "source_reference", "locator": "Theorem 1.4", "context": "proof" }, { "from": "external:Shah_2", "to": "rmk:expanding-curve", "type": "external_reference", "evidence": "f. The ineffective version of Corollary \\ref{cor:curve} was proved in \\cite{Shah_2} using Ratner's theorem. \\subsection*{Acknowledgements} The author thanks Wen Huang, Dmitry Kleinbock, Elon Lindenstrauss, Amir Mohammadi, Nimish Shah, Ralf Spatzier, Zhiren Wang and", "anchor": { "source": "manuscript", "line_start": 601, "line_end": 601, "char_start": 23833, "char_end": 23846 }, "origin": "source_reference", "locator": "", "context": "statement" }, { "from": "external:ratner-acta", "to": "sec-prelim", "type": "external_reference", "evidence": "on we will give the outline of the proof. \\par The proof is inspired by Ratner's original proof of her measure rigidity theorem \\cite{ratner-acta} and recent papers by Lindenstrauss, Mohammadi, and Wang \\cite{LM-preprint2021,LMW-preprint2022}. However, compared with previous works, we take a quite different approach in this paper. \\par In", "anchor": { "source": "manuscript", "line_start": 849, "line_end": 849, "char_start": 36349, "char_end": 36367 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:LM-preprint2021", "to": "sec-prelim", "type": "external_reference", "evidence": "ner's original proof of her measure rigidity theorem \\cite{ratner-acta} and recent papers by Lindenstrauss, Mohammadi, and Wang \\cite{LM-preprint2021,LMW-preprint2022}. However, compared with previous works, we take a quite different approach in this paper. \\par In \\S \\ref{sec-initial-dim-control}, we We will start with $\\cF_0 = a(s)u([1])x_0$ for $s = t - \\delt", "anchor": { "source": "manuscript", "line_start": 850, "line_end": 850, "char_start": 36426, "char_end": 36465 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:LMW-preprint2022", "to": "sec-prelim", "type": "external_reference", "evidence": "ner's original proof of her measure rigidity theorem \\cite{ratner-acta} and recent papers by Lindenstrauss, Mohammadi, and Wang \\cite{LM-preprint2021,LMW-preprint2022}. However, compared with previous works, we take a quite different approach in this paper. \\par In \\S \\ref{sec-initial-dim-control}, we We will start with $\\cF_0 = a(s)u([1])x_0$ for $s = t - \\delt", "anchor": { "source": "manuscript", "line_start": 850, "line_end": 850, "char_start": 36426, "char_end": 36465 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:LM-preprint2021", "to": "sec-prelim", "type": "external_reference", "evidence": "gence} are needed. This is the starting point of our proof. The argument in this part is similar to the corresponding parts in \\cite{LM-preprint2021,LMW-preprint2022}. \\par \\S \\ref{sec-reduction} to \\S \\ref{sec-structured-component} is the crucial part. In these sections, we will construct a sequence $\\{\\cF_i : i \\in \\N\\}$ of $U$-orbits starting from $\\cF_0$", "anchor": { "source": "manuscript", "line_start": 858, "line_end": 858, "char_start": 37318, "char_end": 37357 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:LMW-preprint2022", "to": "sec-prelim", "type": "external_reference", "evidence": "gence} are needed. This is the starting point of our proof. The argument in this part is similar to the corresponding parts in \\cite{LM-preprint2021,LMW-preprint2022}. \\par \\S \\ref{sec-reduction} to \\S \\ref{sec-structured-component} is the crucial part. In these sections, we will construct a sequence $\\{\\cF_i : i \\in \\N\\}$ of $U$-orbits starting from $\\cF_0$", "anchor": { "source": "manuscript", "line_start": 858, "line_end": 858, "char_start": 37318, "char_end": 37357 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:Venkatesh2010", "to": "sec-high-dim-to-equidistribution", "type": "external_reference", "evidence": "s to get effective equidistribution from high transversal dimension control. It can be proved by following Venkatesh's argument (\\cite[\\S 3]{Venkatesh2010} and \\cite[Proposition 4.2]{LM-preprint2021} for details). There exists $ \\theta >0$ such that the following holds: For any $0 < \\e", "anchor": { "source": "manuscript", "line_start": 889, "line_end": 889, "char_start": 40268, "char_end": 40294 }, "origin": "source_reference", "locator": "\\S 3", "context": "narrative" }, { "from": "external:LM-preprint2021", "to": "sec-high-dim-to-equidistribution", "type": "external_reference", "evidence": "tion from high transversal dimension control. It can be proved by following Venkatesh's argument (\\cite[\\S 3]{Venkatesh2010} and \\cite[Proposition 4.2]{LM-preprint2021} for details). There exists $ \\theta >0$ such that the following holds: For any $0 < \\epsilon < 10^{-1000}$, there exist $ \\eta >0$", "anchor": { "source": "manuscript", "line_start": 889, "line_end": 889, "char_start": 40299, "char_end": 40338 }, "origin": "source_reference", "locator": "Proposition 4.2", "context": "narrative" }, { "from": "external:LM-preprint2021", "to": "prop:high-dimension-to-equidistribution-a", "type": "external_reference", "evidence": "\\mu_G \\right| \\ll e^{-\\eta s'} \\|f\\|_S.\\] \\par The statement can be proved by following the proof of \\cite[Proposition 4.2]{LM-preprint2021} step by step.", "anchor": { "source": "manuscript", "line_start": 903, "line_end": 903, "char_start": 41257, "char_end": 41296 }, "origin": "source_reference", "locator": "Proposition 4.2", "context": "proof" }, { "from": "external:kleinbock2008", "to": "prop:quantitative-nondivergence", "type": "external_reference", "evidence": "nd{prop} \\par Let us denote $x = g \\Gamma$. By the Kleinbock-Margulis quantitative non-divergence theorem (cf. \\cite[Theorem 2.2]{kleinbock2008}, \\cite[Theorem 5.2]{Klein_Mar}), if the statement does not hold then there exists a nonzero $\\vv v \\in \\bigwedge^j \\Z^3$ (where $j = 1$ or $2$), such that \\[ \\max_{r \\in [1]}\\| a(s) u(r) g \\vv", "anchor": { "source": "manuscript", "line_start": 927, "line_end": 927, "char_start": 42616, "char_end": 42649 }, "origin": "source_reference", "locator": "Theorem 2.2", "context": "proof" }, { "from": "external:Klein_Mar", "to": "prop:quantitative-nondivergence", "type": "external_reference", "evidence": "s denote $x = g \\Gamma$. By the Kleinbock-Margulis quantitative non-divergence theorem (cf. \\cite[Theorem 2.2]{kleinbock2008}, \\cite[Theorem 5.2]{Klein_Mar}), if the statement does not hold then there exists a nonzero $\\vv v \\in \\bigwedge^j \\Z^3$ (where $j = 1$ or $2$), such that \\[ \\max_{r \\in [1]}\\| a(s) u(r) g \\vv v\\| \\le 1. \\] \\par \\textbf{Ca", "anchor": { "source": "manuscript", "line_start": 927, "line_end": 927, "char_start": 42651, "char_end": 42680 }, "origin": "source_reference", "locator": "Theorem 5.2", "context": "proof" }, { "from": "external:einsiedler-margulis-venkatesh", "to": "prop:initial-bound-dimension", "type": "external_reference", "evidence": "in \\SL(3,\\Z)$, if $\\gamma_{\\bar x}$ has nontrivial unipotent part, it must be unipotent. By repeating the same argument as in \\cite{einsiedler-margulis-venkatesh}, we can find $g' \\in G$ satisfying that $\\gamma_{\\bar x} g'^{-1} v_H = g'^{-1} v_H$ and \\[ \\|g' - g\\| \\le \\| \\gamma_{\\bar x} v_0 - v_0\\| \\|\\gamma_{\\bar x}\\|^{10} \\le e^{- (1-\\eta_1)\\ell} e^{10 \\et", "anchor": { "source": "manuscript", "line_start": 1134, "line_end": 1134, "char_start": 55970, "char_end": 56006 }, "origin": "source_reference", "locator": "", "context": "proof" }, { "from": "external:einsiedler-margulis-venkatesh", "to": "prop:initial-bound-dimension", "type": "external_reference", "evidence": "x}$ and $\\gamma' = \\gamma_{\\bar x'}$ not commuting. Using the same argument as in Case 1 (again using the argument in \\cite{einsiedler-margulis-venkatesh}), we can find we can find $g' \\in G$ satisfying that $\\gamma g'^{-1} v_H = g'^{-1} v_H$, $\\gamma' g'^{-1} v_H = g'^{-1} v_H$ and \\[ \\|g' - g\\| \\le e^{-\\ell/2}. \\] This implies that $g' \\ga", "anchor": { "source": "manuscript", "line_start": 1158, "line_end": 1158, "char_start": 58228, "char_end": 58264 }, "origin": "source_reference", "locator": "", "context": "proof" }, { "from": "external:ratner_2", "to": "thm:ratner", "type": "background", "evidence": "In 1991, Ratner \\cite{ratner_2} proved the following fundamental theorem on", "anchor": { "source": "manuscript", "line_start": 381, "line_end": 381, "char_start": 10047, "char_end": 10122 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "In 1991, Ratner \\cite{ratner_2} proved the following fundamental theorem on", "locator": "", "audit_status": "not_inspected", "imported_result": "", "required_hypotheses": [], "reference_role": "background" }, { "from": "thm:main-thm", "to": "thm:effective-U-orbit", "type": "proof_dependency", "evidence": "Proving Theorem \\ref{thm:effective-U-orbit} is equivalent to proving the following theorem:", "anchor": { "source": "manuscript", "line_start": 489, "line_end": 490, "char_start": 17808, "char_end": 17899 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "Proving Theorem \\ref{thm:effective-U-orbit} is equivalent to proving the following theorem:" }, { "from": "prop:initial-bound-dimension", "to": "thm:main-thm", "type": "proof_dependency", "evidence": "By Proposition \\ref{prop:initial-bound-dimension}, we have that $\\mu_{\\cF_0}$ is $(d_0, s_0)$-good.", "anchor": { "source": "manuscript", "line_start": 3342, "line_end": 3342, "char_start": 202875, "char_end": 202974 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Proposition \\ref{prop:initial-bound-dimension}, we have that $\\mu_{\\cF_0}$ is $(d_0, s_0)$-good." }, { "from": "prop:improve-dimension", "to": "thm:main-thm", "type": "proof_dependency", "evidence": "By Proposition \\ref{prop:dimension-improvement-random}, \\ref{prop:dimension-improvement-strange-part}, and \\ref{prop:dimension-improvement-structured-component} we get Proposition \\ref{prop:improve-dimension}.", "anchor": { "source": "manuscript", "line_start": 3342, "line_end": 3342, "char_start": 202975, "char_end": 203184 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Proposition \\ref{prop:dimension-improvement-random}, \\ref{prop:dimension-improvement-strange-part}, and \\ref{prop:dimension-improvement-structured-component} we get Proposition \\ref{prop:improve-dimension}." }, { "from": "prop:reduction", "to": "thm:main-thm", "type": "proof_dependency", "evidence": "By Proposition \\ref{prop:reduction}, it implies that $\\mu_{\\cF_{i}}$ is $(d_i, s_i)$-good, where $d_i = d_0 + i \\epsilon_2$ and $s_i = 2^{-i}s_0$(we remove an exponentially small proportion at each step).", "anchor": { "source": "manuscript", "line_start": 3342, "line_end": 3342, "char_start": 203185, "char_end": 203389 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Proposition \\ref{prop:reduction}, it implies that $\\mu_{\\cF_{i}}$ is $(d_i, s_i)$-good, where $d_i = d_0 + i \\epsilon_2$ and $s_i = 2^{-i}s_0$(we remove an exponentially small proportion at each step)." }, { "from": "prop:high-dim-decomposition", "to": "thm:main-thm", "type": "proof_dependency", "evidence": "By Proposition \\ref{prop:high-dim-decomposition}, we have that", "anchor": { "source": "manuscript", "line_start": 3343, "line_end": 3343, "char_start": 203468, "char_end": 203530 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Proposition \\ref{prop:high-dim-decomposition}, we have that" }, { "from": "prop:high-dimension-to-equidistribution-a", "to": "thm:main-thm", "type": "proof_dependency", "evidence": "Then Proposition \\ref{prop:high-dimension-to-equidistribution-a}, we have \n that $a(2 s_i)_\\ast \\mu_{k}$ is effectively equidistributed.", "anchor": { "source": "manuscript", "line_start": 3347, "line_end": 3348, "char_start": 203788, "char_end": 203924 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "Then Proposition \\ref{prop:high-dimension-to-equidistribution-a}, we have \n that $a(2 s_i)_\\ast \\mu_{k}$ is effectively equidistributed." }, { "from": "thm:main-thm", "to": "cor:straight-line", "type": "proof_dependency", "evidence": "Applying Theorem \\ref{thm:main-thm} with $x_0 = a_0(t) n^\\ast(-w_1, w_2)\\Gamma$ and $f_{w_1}$, we have that", "anchor": { "source": "manuscript", "line_start": 543, "line_end": 543, "char_start": 20456, "char_end": 20563 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "Applying Theorem \\ref{thm:main-thm} with $x_0 = a_0(t) n^\\ast(-w_1, w_2)\\Gamma$ and $f_{w_1}$, we have that" }, { "from": "external:KNSY", "to": "cor:straight-line", "type": "background", "evidence": "The qualitative version of Corollary \\ref{cor:straight-line} was proved in \\cite{KNSY} for $\\omega(\\vv w) \\le 2$ using Ratner's theorem.", "anchor": { "source": "manuscript", "line_start": 550, "line_end": 550, "char_start": 20812, "char_end": 20948 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "The qualitative version of Corollary \\ref{cor:straight-line} was proved in \\cite{KNSY} for $\\omega(\\vv w) \\le 2$ using Ratner's theorem.", "locator": "", "audit_status": "not_inspected", "imported_result": "", "required_hypotheses": [], "reference_role": "background" }, { "from": "thm:main-thm", "to": "cor:curve", "type": "proof_dependency", "evidence": "By Theorem \\ref{thm:main-thm}, if we have that for any", "anchor": { "source": "manuscript", "line_start": 582, "line_end": 582, "char_start": 22632, "char_end": 22686 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Theorem \\ref{thm:main-thm}, if we have that for any" }, { "from": "external:Bernik-Kleinbock-Margulis", "to": "cor:curve", "type": "proof_dependency", "evidence": "By \\cite[Theorem 1.4]{Bernik-Kleinbock-Margulis}, we have that", "anchor": { "source": "manuscript", "line_start": 587, "line_end": 587, "char_start": 23055, "char_end": 23117 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By \\cite[Theorem 1.4]{Bernik-Kleinbock-Margulis}, we have that", "locator": "Theorem 1.4", "audit_status": "not_inspected", "imported_result": "Theorem 1.4, invoked to obtain a uniform exponentially small bound for Leb(fm_ell). The exact external statement is not supplied.", "required_hypotheses": [ "The parameter-dependent lattice map must satisfy the theorem's quantitative goodness and nondegeneracy hypotheses.", "The required lower bounds on the relevant covolume functions must hold uniformly over the stated ell range.", "The theorem must control the compactness condition involving both vectors and exterior-square vectors." ], "reference_role": "proof_input" }, { "from": "external:Shah_2", "to": "cor:curve", "type": "background", "evidence": "The ineffective version of Corollary \\ref{cor:curve} was proved in \\cite{Shah_2} using Ratner's theorem.", "anchor": { "source": "manuscript", "line_start": 601, "line_end": 602, "char_start": 23766, "char_end": 23870 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "The ineffective version of Corollary \\ref{cor:curve} was proved in \\cite{Shah_2} using Ratner's theorem.", "locator": "", "audit_status": "not_inspected", "imported_result": "", "required_hypotheses": [], "reference_role": "background" }, { "from": "lm:lie-bracket", "to": "lm:product-lie-algebra-1", "type": "proof_dependency", "evidence": "By Lemma \\ref{lm:lie-bracket} and the fact that for any $\\vv w_1, \\vv w_2 \\in \\fg$,", "anchor": { "source": "manuscript", "line_start": 779, "line_end": 779, "char_start": 31605, "char_end": 31688 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Lemma \\ref{lm:lie-bracket} and the fact that for any $\\vv w_1, \\vv w_2 \\in \\fg$," }, { "from": "lm:lie-bracket", "to": "lm:product-w", "type": "proof_dependency", "evidence": "By Lemma \\ref{lm:lie-bracket}, we have the following observations:", "anchor": { "source": "manuscript", "line_start": 812, "line_end": 813, "char_start": 34097, "char_end": 34163 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Lemma \\ref{lm:lie-bracket}, we have the following observations:" }, { "from": "lm:product-lie-algebra-1", "to": "lm:product-w", "type": "proof_dependency", "evidence": "By Lemma \\ref{lm:product-lie-algebra-1}, we have that", "anchor": { "source": "manuscript", "line_start": 833, "line_end": 833, "char_start": 35555, "char_end": 35608 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Lemma \\ref{lm:product-lie-algebra-1}, we have that" }, { "from": "external:LM-preprint2021", "to": "prop:high-dimension-to-equidistribution-a", "type": "proof_dependency", "evidence": "The statement can be proved by following the proof of \\cite[Proposition 4.2]{LM-preprint2021} step by step.", "anchor": { "source": "manuscript", "line_start": 903, "line_end": 904, "char_start": 41203, "char_end": 41310 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "The statement can be proved by following the proof of \\cite[Proposition 4.2]{LM-preprint2021} step by step.", "locator": "Proposition 4.2", "audit_status": "not_inspected", "imported_result": "Proposition 4.2 and its proof, claimed to transfer step by step to the present SL(3,R) action. The exact external theorem and the transfer are not supplied.", "required_hypotheses": [ "A suitable quantitative matrix-coefficient estimate for the present G action and Sobolev norm.", "The stated Frostman bound at scale beta e^{-s'}.", "Uniform control for base points in X_{beta^{1/4}}.", "Verification that the transverse directions, flow relations and parameter ranges in the cited proof apply here." ], "reference_role": "proof_input" }, { "from": "external:Venkatesh2010", "to": "prop:high-dimension-to-equidistribution-a", "type": "proof_dependency", "evidence": "It can be proved by following Venkatesh's argument (\\cite[\\S 3]{Venkatesh2010} and \\cite[Proposition 4.2]{LM-preprint2021} for details).", "anchor": { "source": "manuscript", "line_start": 889, "line_end": 890, "char_start": 40216, "char_end": 40352 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "It can be proved by following Venkatesh's argument (\\cite[\\S 3]{Venkatesh2010} and \\cite[Proposition 4.2]{LM-preprint2021} for details).", "locator": "Section 3", "audit_status": "not_inspected", "imported_result": "The van der Corput-type equidistribution argument in Section 3, cited as the method underlying this proposition. No exact transferable theorem is identified.", "required_hypotheses": [ "Quantitative mixing and the regularity bounds needed by the averaging argument.", "Control of correlations for the particular transverse measure and unipotent subgroup used here." ], "reference_role": "proof_input" }, { "from": "external:kleinbock2008", "to": "prop:quantitative-nondivergence", "type": "proof_dependency", "evidence": "By the Kleinbock-Margulis quantitative non-divergence theorem \n (cf. \\cite[Theorem 2.2]{kleinbock2008}, \\cite[Theorem 5.2]{Klein_Mar}), if the statement does not hold then there exists", "anchor": { "source": "manuscript", "line_start": 926, "line_end": 927, "char_start": 42546, "char_end": 42731 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By the Kleinbock-Margulis quantitative non-divergence theorem \n (cf. \\cite[Theorem 2.2]{kleinbock2008}, \\cite[Theorem 5.2]{Klein_Mar}), if the statement does not hold then there exists", "locator": "Theorem 2.2", "audit_status": "not_inspected", "imported_result": "Theorem 2.2, invoked as quantitative non-divergence for the polynomial lattice map r -> a(s)u(r)g.", "required_hypotheses": [ "Uniform goodness of the relevant exterior-power covolume functions.", "A positive uniform lower bound for their suprema on the parameter interval.", "The theorem's lattice-rank and interval hypotheses." ], "reference_role": "proof_input" }, { "from": "external:Klein_Mar", "to": "prop:quantitative-nondivergence", "type": "proof_dependency", "evidence": "By the Kleinbock-Margulis quantitative non-divergence theorem \n (cf. \\cite[Theorem 2.2]{kleinbock2008}, \\cite[Theorem 5.2]{Klein_Mar}), if the statement does not hold then there exists", "anchor": { "source": "manuscript", "line_start": 926, "line_end": 927, "char_start": 42546, "char_end": 42731 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By the Kleinbock-Margulis quantitative non-divergence theorem \n (cf. \\cite[Theorem 2.2]{kleinbock2008}, \\cite[Theorem 5.2]{Klein_Mar}), if the statement does not hold then there exists", "locator": "Theorem 5.2", "audit_status": "not_inspected", "imported_result": "Theorem 5.2, cited as an alternative quantitative non-divergence input.", "required_hypotheses": [ "Goodness of all relevant primitive-subgroup covolume functions.", "Uniform lower bounds on their suprema.", "The applicable interval and lattice hypotheses." ], "reference_role": "proof_input" }, { "from": "lm:separation-lemma-a", "to": "prop:initial-bound-dimension", "type": "proof_dependency", "evidence": "by Lemma \\ref{lm:separation-lemma-a}, if we write $h_{\\bar x} = \\exp(\\vv h_{\\bar x})$, and", "anchor": { "source": "manuscript", "line_start": 1070, "line_end": 1070, "char_start": 51166, "char_end": 51256 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "by Lemma \\ref{lm:separation-lemma-a}, if we write $h_{\\bar x} = \\exp(\\vv h_{\\bar x})$, and" }, { "from": "external:einsiedler-margulis-venkatesh", "to": "prop:initial-bound-dimension", "type": "proof_dependency", "evidence": "By repeating the same argument as in \\cite{einsiedler-margulis-venkatesh},", "anchor": { "source": "manuscript", "line_start": 1134, "line_end": 1135, "char_start": 55933, "char_end": 56007 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By repeating the same argument as in \\cite{einsiedler-margulis-venkatesh},", "locator": "", "audit_status": "not_inspected", "imported_result": "An effective approximate-stabilizer-to-exact-stabilizer argument, used first for one integer matrix and then simultaneously for two noncommuting matrices. No theorem or lemma locator is supplied.", "required_hypotheses": [ "A specified representation and algebraic stabilizer.", "Height bounds for the integer matrices and the representative g.", "A quantitative distance-to-exact-stabilizer estimate with the stated exponents.", "For Case 2, a simultaneous estimate for both matrices." ], "reference_role": "proof_input" }, { "from": "prop:initial-bound-dimension", "to": "prop:sl2xr2-closing-lemma", "type": "proof_dependency", "evidence": "By repeating the argument in the proof of \n Proposition \\ref{prop:initial-bound-dimension}, we can show that", "anchor": { "source": "manuscript", "line_start": 1184, "line_end": 1185, "char_start": 59860, "char_end": 59968 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By repeating the argument in the proof of \n Proposition \\ref{prop:initial-bound-dimension}, we can show that" }, { "from": "lm:product-w", "to": "lm:two-theta", "type": "proof_dependency", "evidence": "By Lemma \\ref{lm:product-w},", "anchor": { "source": "manuscript", "line_start": 1324, "line_end": 1324, "char_start": 68282, "char_end": 68310 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Lemma \\ref{lm:product-w}," }, { "from": "lm:two-theta", "to": "lm:product-theta", "type": "proof_dependency", "evidence": "By Lemma \\ref{lm:two-theta}, it suffices to show that", "anchor": { "source": "manuscript", "line_start": 1342, "line_end": 1342, "char_start": 69124, "char_end": 69177 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Lemma \\ref{lm:two-theta}, it suffices to show that" }, { "from": "lm:product-w", "to": "lm:product-theta", "type": "proof_dependency", "evidence": "By Lemma \\ref{lm:product-w}, we have", "anchor": { "source": "manuscript", "line_start": 1347, "line_end": 1347, "char_start": 69527, "char_end": 69563 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Lemma \\ref{lm:product-w}, we have" }, { "from": "lm:product-theta", "to": "cor:product-theta", "type": "proof_dependency", "evidence": "By repeating the proof of Lemma \\ref{lm:product-theta}, we get", "anchor": { "source": "manuscript", "line_start": 1367, "line_end": 1367, "char_start": 70745, "char_end": 70807 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By repeating the proof of Lemma \\ref{lm:product-theta}, we get" }, { "from": "lm:product-w", "to": "lm:different-coordinate-1", "type": "proof_dependency", "evidence": "by Lemma \\ref{lm:product-w}, we have", "anchor": { "source": "manuscript", "line_start": 1392, "line_end": 1392, "char_start": 72669, "char_end": 72705 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "by Lemma \\ref{lm:product-w}, we have" }, { "from": "eq:fy", "to": "lm:cross-section", "type": "proof_dependency", "evidence": "Moreover, by \\eqref{eq:fy}, we have", "anchor": { "source": "manuscript", "line_start": 1549, "line_end": 1549, "char_start": 82975, "char_end": 83010 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "Moreover, by \\eqref{eq:fy}, we have" }, { "from": "lm:different-coordinate-1", "to": "lm:initial-dimension-under-a-t", "type": "proof_dependency", "evidence": "By Lemma \\ref{lm:different-coordinate-1}, we have that", "anchor": { "source": "manuscript", "line_start": 1442, "line_end": 1442, "char_start": 75948, "char_end": 76002 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Lemma \\ref{lm:different-coordinate-1}, we have that" }, { "from": "prop:initial-bound-dimension", "to": "lm:initial-dimension-under-a-t", "type": "proof_dependency", "evidence": "By repeating the proof of Proposition \\ref{prop:initial-bound-dimension} to $\\cF_i$ we have", "anchor": { "source": "manuscript", "line_start": 1444, "line_end": 1444, "char_start": 76201, "char_end": 76292 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By repeating the proof of Proposition \\ref{prop:initial-bound-dimension} to $\\cF_i$ we have" }, { "from": "lm:product-theta", "to": "lm:partition-1", "type": "proof_dependency", "evidence": "By Lemma \\ref{lm:product-theta}, we have", "anchor": { "source": "manuscript", "line_start": 1642, "line_end": 1642, "char_start": 89453, "char_end": 89493 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Lemma \\ref{lm:product-theta}, we have" }, { "from": "lm:different-coordinate-1", "to": "lm:partition-1", "type": "proof_dependency", "evidence": "By Lemma \\ref{lm:different-coordinate-1}, we have", "anchor": { "source": "manuscript", "line_start": 1442, "line_end": 1442, "char_start": 75948, "char_end": 75997 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Lemma \\ref{lm:different-coordinate-1}, we have" }, { "from": "lm:two-theta", "to": "lm:partition-2", "type": "proof_dependency", "evidence": "By Lemma \\ref{lm:two-theta}, $\\Theta_{s'} \\subset \\hat{\\Theta}_{s'}$.", "anchor": { "source": "manuscript", "line_start": 1677, "line_end": 1677, "char_start": 91938, "char_end": 92007 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Lemma \\ref{lm:two-theta}, $\\Theta_{s'} \\subset \\hat{\\Theta}_{s'}$." }, { "from": "lm:product-w", "to": "lm:partition-2", "type": "proof_dependency", "evidence": "By Lemma \\ref{lm:product-w}, we have that", "anchor": { "source": "manuscript", "line_start": 1688, "line_end": 1688, "char_start": 92631, "char_end": 92672 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Lemma \\ref{lm:product-w}, we have that" }, { "from": "lm:different-coordinate-1", "to": "lm:partition-2", "type": "proof_dependency", "evidence": "By Lemma \\ref{lm:different-coordinate-1},", "anchor": { "source": "manuscript", "line_start": 1442, "line_end": 1442, "char_start": 75948, "char_end": 75989 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Lemma \\ref{lm:different-coordinate-1}," }, { "from": "lm:cross-section", "to": "lm:omega-u", "type": "proof_dependency", "evidence": "By Lemma \\ref{lm:cross-section}, $u(\\fl(y'))y' \\cap \\Sigma^{\\tilde{r}}$ is contained in an interval of length $\\asymp \\beta$.", "anchor": { "source": "manuscript", "line_start": 1794, "line_end": 1794, "char_start": 98745, "char_end": 98870 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Lemma \\ref{lm:cross-section}, $u(\\fl(y'))y' \\cap \\Sigma^{\\tilde{r}}$ is contained in an interval of length $\\asymp \\beta$." }, { "from": "lm:omega-u", "to": "lm:time-leave", "type": "proof_dependency", "evidence": "By Lemma \\ref{lm:omega-u}, we have $\\Omega(y_i, L_i) \\subset \\hat{\\Omega}(\\tilde{y}_i)$.", "anchor": { "source": "manuscript", "line_start": 1823, "line_end": 1823, "char_start": 100576, "char_end": 100664 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Lemma \\ref{lm:omega-u}, we have $\\Omega(y_i, L_i) \\subset \\hat{\\Omega}(\\tilde{y}_i)$." }, { "from": "cor:product-theta", "to": "lm:time-leave", "type": "proof_dependency", "evidence": "By Corollary \\ref{cor:product-theta},", "anchor": { "source": "manuscript", "line_start": 1829, "line_end": 1830, "char_start": 101163, "char_end": 101200 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Corollary \\ref{cor:product-theta}," }, { "from": "lm:different-coordinate-1", "to": "lm:time-leave", "type": "proof_dependency", "evidence": "By Lemma \\ref{lm:different-coordinate-1}, if we write", "anchor": { "source": "manuscript", "line_start": 1833, "line_end": 1833, "char_start": 101456, "char_end": 101509 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Lemma \\ref{lm:different-coordinate-1}, if we write" }, { "from": "lm:time-leave", "to": "lm:concentrated-surface", "type": "proof_dependency", "evidence": "By Lemma \\ref{lm:time-leave}, $ r \\in I(\\bar z, y)$ implies that $|f_v(r)|, |f_w(r)| \\le 4.6 \\beta e^{-s'}$.", "anchor": { "source": "manuscript", "line_start": 1896, "line_end": 1896, "char_start": 105760, "char_end": 105868 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Lemma \\ref{lm:time-leave}, $ r \\in I(\\bar z, y)$ implies that $|f_v(r)|, |f_w(r)| \\le 4.6 \\beta e^{-s'}$." }, { "from": "prop:dimension-improvement-random", "to": "prop:improve-dimension", "type": "proof_dependency", "evidence": "By Proposition \\ref{prop:dimension-improvement-random}, \\ref{prop:dimension-improvement-strange-part}, and \\ref{prop:dimension-improvement-structured-component} we get Proposition \\ref{prop:improve-dimension}.", "anchor": { "source": "manuscript", "line_start": 3342, "line_end": 3342, "char_start": 202975, "char_end": 203184 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Proposition \\ref{prop:dimension-improvement-random}, \\ref{prop:dimension-improvement-strange-part}, and \\ref{prop:dimension-improvement-structured-component} we get Proposition \\ref{prop:improve-dimension}." }, { "from": "prop:dimension-improvement-strange-part", "to": "prop:improve-dimension", "type": "proof_dependency", "evidence": "By Proposition \\ref{prop:dimension-improvement-random}, \\ref{prop:dimension-improvement-strange-part}, and \\ref{prop:dimension-improvement-structured-component} we get Proposition \\ref{prop:improve-dimension}.", "anchor": { "source": "manuscript", "line_start": 3342, "line_end": 3342, "char_start": 202975, "char_end": 203184 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Proposition \\ref{prop:dimension-improvement-random}, \\ref{prop:dimension-improvement-strange-part}, and \\ref{prop:dimension-improvement-structured-component} we get Proposition \\ref{prop:improve-dimension}." }, { "from": "prop:dimension-improvement-structured-component", "to": "prop:improve-dimension", "type": "proof_dependency", "evidence": "By Proposition \\ref{prop:dimension-improvement-random}, \\ref{prop:dimension-improvement-strange-part}, and \\ref{prop:dimension-improvement-structured-component} we get Proposition \\ref{prop:improve-dimension}.", "anchor": { "source": "manuscript", "line_start": 3342, "line_end": 3342, "char_start": 202975, "char_end": 203184 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Proposition \\ref{prop:dimension-improvement-random}, \\ref{prop:dimension-improvement-strange-part}, and \\ref{prop:dimension-improvement-structured-component} we get Proposition \\ref{prop:improve-dimension}." }, { "from": "lm:time-leave", "to": "prop:dimension-improvement-random", "type": "proof_dependency", "evidence": "By \\eqref{eq:assumption-1} and Lemma \\ref{lm:time-leave},", "anchor": { "source": "manuscript", "line_start": 2168, "line_end": 2168, "char_start": 122495, "char_end": 122552 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By \\eqref{eq:assumption-1} and Lemma \\ref{lm:time-leave}," }, { "from": "lm:concentrated-surface", "to": "prop:dimension-improvement-random", "type": "proof_dependency", "evidence": "By Lemma \\ref{lm:concentrated-surface}, $\\fD_{z}(z') \\in \\cC(\\cT(y), z)$.", "anchor": { "source": "manuscript", "line_start": 2177, "line_end": 2177, "char_start": 123887, "char_end": 123960 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Lemma \\ref{lm:concentrated-surface}, $\\fD_{z}(z') \\in \\cC(\\cT(y), z)$." }, { "from": "prop:dimension-improvement-random", "to": "prop:dimension-improvement-strange-part", "type": "proof_dependency", "evidence": "Using similar arguments as in the proof of Proposition \\ref{prop:dimension-improvement-random}, we can get", "anchor": { "source": "manuscript", "line_start": 2229, "line_end": 2229, "char_start": 127649, "char_end": 127755 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "Using similar arguments as in the proof of Proposition \\ref{prop:dimension-improvement-random}, we can get" }, { "from": "lm:time-leave", "to": "prop:dimension-improvement-strange-part", "type": "proof_dependency", "evidence": "Since $\\|\\Delta(y, y_2)\\| \\ge \\beta e^{-s' - j-1}$, by Lemma \\ref{lm:time-leave}, for each $y_2 \\in \\fJ\\cap \\Sigma^r$ such that $(y, y_2) \\in \\diamond_{r,j}$, we have that $I(y, y_2) \\subset I$ where $I$ is an interval of length $\\le e^{j+1}$.", "anchor": { "source": "manuscript", "line_start": 2272, "line_end": 2272, "char_start": 130916, "char_end": 131160 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "Since $\\|\\Delta(y, y_2)\\| \\ge \\beta e^{-s' - j-1}$, by Lemma \\ref{lm:time-leave}, for each $y_2 \\in \\fJ\\cap \\Sigma^r$ such that $(y, y_2) \\in \\diamond_{r,j}$, we have that $I(y, y_2) \\subset I$ where $I$ is an interval of length $\\le e^{j+1}$." }, { "from": "lem:sigma j lemma", "to": "prop:dimension-improvement-strange-part", "type": "proof_dependency", "evidence": "Using similar arguments to those used in the proof of Lemma~\\ref{lem:sigma j lemma}, we can prove that", "anchor": { "source": "manuscript", "line_start": 2330, "line_end": 2330, "char_start": 135427, "char_end": 135529 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "Using similar arguments to those used in the proof of Lemma~\\ref{lem:sigma j lemma}, we can prove that" }, { "from": "lm:initial-dimension-under-a-t", "to": "prop:dimension-improvement-strange-part", "type": "proof_dependency", "evidence": "By Lemma \\ref{lm:initial-dimension-under-a-t}, we have", "anchor": { "source": "manuscript", "line_start": 2349, "line_end": 2349, "char_start": 136563, "char_end": 136617 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Lemma \\ref{lm:initial-dimension-under-a-t}, we have" }, { "from": "lm:concentrated-surface", "to": "prop:dimension-improvement-strange-part", "type": "proof_dependency", "evidence": "By Lemma \\ref{lm:concentrated-surface}, the coordinates $\\fD_{ z}(y_1)$ of all $y_1\\in \\cI_z$ such that $(y_2,y_1) \\in \\diamond_{r,j}$", "anchor": { "source": "manuscript", "line_start": 2339, "line_end": 2340, "char_start": 135899, "char_end": 136033 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Lemma \\ref{lm:concentrated-surface}, the coordinates $\\fD_{ z}(y_1)$ of all $y_1\\in \\cI_z$ such that $(y_2,y_1) \\in \\diamond_{r,j}$" }, { "from": "lm:concentrated-surface", "to": "lem:sigma j lemma", "type": "proof_dependency", "evidence": "Then by Lemma~\\ref{lm:concentrated-surface}, we have $\\|\\vv w^+\\| \\le \\beta e^{s' - j}$.", "anchor": { "source": "manuscript", "line_start": 2303, "line_end": 2303, "char_start": 133222, "char_end": 133310 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "Then by Lemma~\\ref{lm:concentrated-surface}, we have $\\|\\vv w^+\\| \\le \\beta e^{s' - j}$." }, { "from": "lm:product-w", "to": "lem:sigma j lemma", "type": "proof_dependency", "evidence": "By Lemma \\ref{lm:product-w},", "anchor": { "source": "manuscript", "line_start": 1324, "line_end": 1324, "char_start": 68282, "char_end": 68310 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Lemma \\ref{lm:product-w}," }, { "from": "prop:dimension-improvement-random", "to": "prop:dimension-does-not-drop", "type": "proof_dependency", "evidence": "which follows the proof of Proposition \\ref{prop:dimension-improvement-random}.", "anchor": { "source": "manuscript", "line_start": 2410, "line_end": 2411, "char_start": 140752, "char_end": 140831 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "which follows the proof of Proposition \\ref{prop:dimension-improvement-random}." }, { "from": "prop:dimension-improvement-strange-part", "to": "prop:dimension-does-not-drop", "type": "proof_dependency", "evidence": "which follows the proof of Proposition \\ref{prop:dimension-improvement-strange-part}.", "anchor": { "source": "manuscript", "line_start": 2469, "line_end": 2470, "char_start": 144966, "char_end": 145051 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "which follows the proof of Proposition \\ref{prop:dimension-improvement-strange-part}." }, { "from": "lm:concentrated-surface", "to": "prop:dimension-does-not-drop", "type": "proof_dependency", "evidence": "For the upper bound, by Lemma \\ref{lm:concentrated-surface}, for any $y_1 \\in \\fJ \\cap \\Sigma^r$ and $z \\in \\cI \\cap \\Sigma^r \\cap \\fZ'$, we have that for any $y_2 \\in \\cI_z$ such that $(y_1, y_2) \\in \\diamond'_r$, we have", "anchor": { "source": "manuscript", "line_start": 2448, "line_end": 2448, "char_start": 143566, "char_end": 143788 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "For the upper bound, by Lemma \\ref{lm:concentrated-surface}, for any $y_1 \\in \\fJ \\cap \\Sigma^r$ and $z \\in \\cI \\cap \\Sigma^r \\cap \\fZ'$, we have that for any $y_2 \\in \\cI_z$ such that $(y_1, y_2) \\in \\diamond'_r$, we have" }, { "from": "prop:dimension-does-not-drop", "to": "prop:dimension-improvement-structured-component", "type": "proof_dependency", "evidence": "Let $\\fZ \\subset \\Pi(x)$ be the subset from Proposition \\ref{prop:dimension-does-not-drop}.", "anchor": { "source": "manuscript", "line_start": 2940, "line_end": 2940, "char_start": 174253, "char_end": 174344 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "Let $\\fZ \\subset \\Pi(x)$ be the subset from Proposition \\ref{prop:dimension-does-not-drop}." }, { "from": "lm:integrable-surface", "to": "prop:dimension-improvement-structured-component", "type": "proof_dependency", "evidence": "By Lemma \\ref{lm:integrable-surface}, we have that for every highly concentrated curve neighborhood $\\cC$, there exist at most", "anchor": { "source": "manuscript", "line_start": 3066, "line_end": 3066, "char_start": 183453, "char_end": 183579 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Lemma \\ref{lm:integrable-surface}, we have that for every highly concentrated curve neighborhood $\\cC$, there exist at most" }, { "from": "lm:determine-curve-nbhd", "to": "prop:dimension-improvement-structured-component", "type": "proof_dependency", "evidence": "By Lemma \\ref{lm:determine-curve-nbhd}, we have that the curve neighborhoods from $\\mathfrak{L}_n(\\cC, m_{\\cC})$ are contained", "anchor": { "source": "manuscript", "line_start": 3118, "line_end": 3118, "char_start": 187130, "char_end": 187256 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Lemma \\ref{lm:determine-curve-nbhd}, we have that the curve neighborhoods from $\\mathfrak{L}_n(\\cC, m_{\\cC})$ are contained" }, { "from": "lm:counting-curves-nbhd", "to": "prop:dimension-improvement-structured-component", "type": "proof_dependency", "evidence": "By the last statement of Lemma \\ref{lm:counting-curves-nbhd}, we have that each curve neighborhood from $\\mathfrak{L}_n(\\cC, m_{\\cC})$ can cover at most", "anchor": { "source": "manuscript", "line_start": 3121, "line_end": 3121, "char_start": 187492, "char_end": 187644 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By the last statement of Lemma \\ref{lm:counting-curves-nbhd}, we have that each curve neighborhood from $\\mathfrak{L}_n(\\cC, m_{\\cC})$ can cover at most" }, { "from": "lm:curve-in-N", "to": "prop:dimension-improvement-structured-component", "type": "proof_dependency", "evidence": "Indeed, by Lemma \\ref{lm:curve-in-N}, there exists $z' \\in \\Theta'(z)$ so that the following holds:", "anchor": { "source": "manuscript", "line_start": 3040, "line_end": 3040, "char_start": 181250, "char_end": 181349 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "Indeed, by Lemma \\ref{lm:curve-in-N}, there exists $z' \\in \\Theta'(z)$ so that the following holds:" }, { "from": "prop:sl2xr2-closing-lemma", "to": "prop:dimension-improvement-structured-component", "type": "proof_dependency", "evidence": "By our discussion at the beginning of \\S \\ref{sec-reduction}, Proposition \\ref{prop:sl2xr2-closing-lemma} holds for $\\mu_{\\cF_i}$.", "anchor": { "source": "manuscript", "line_start": 3045, "line_end": 3045, "char_start": 181655, "char_end": 181785 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By our discussion at the beginning of \\S \\ref{sec-reduction}, Proposition \\ref{prop:sl2xr2-closing-lemma} holds for $\\mu_{\\cF_i}$." }, { "from": "lm:hyperbola-plane", "to": "lm:integrable-surface", "type": "proof_dependency", "evidence": "By Lemma \\ref{lm:hyperbola-plane}(1) with $A =2 e^{11C_1\\epsilon_1 s'}$, we have that", "anchor": { "source": "manuscript", "line_start": 2752, "line_end": 2752, "char_start": 162512, "char_end": 162597 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Lemma \\ref{lm:hyperbola-plane}(1) with $A =2 e^{11C_1\\epsilon_1 s'}$, we have that" }, { "from": "rmk:concentrated-surface", "to": "lm:integrable-surface", "type": "proof_dependency", "evidence": "In this case, by Remark \\ref{rmk:concentrated-surface}, we can consider the corresponding straight lines", "anchor": { "source": "manuscript", "line_start": 2760, "line_end": 2760, "char_start": 163208, "char_end": 163312 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "In this case, by Remark \\ref{rmk:concentrated-surface}, we can consider the corresponding straight lines" }, { "from": "lm:hyperbola-plane", "to": "lm:determine-curve-nbhd", "type": "proof_dependency", "evidence": "By Lemma \\ref{lm:hyperbola-plane}(2) with $A = e^{2s' - 2\\ell -m}$ we have that $\\cC'$ is contained in the $e^{-s' - m - 2\\ell}$-neighborhood of $\\cC$.", "anchor": { "source": "manuscript", "line_start": 2802, "line_end": 2802, "char_start": 165921, "char_end": 166072 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Lemma \\ref{lm:hyperbola-plane}(2) with $A = e^{2s' - 2\\ell -m}$ we have that $\\cC'$ is contained in the $e^{-s' - m - 2\\ell}$-neighborhood of $\\cC$." }, { "from": "lm:hyperbola-plane", "to": "lm:counting-curves-nbhd", "type": "proof_dependency", "evidence": "By Lemma \\ref{lm:hyperbola-plane} (1) with $A = e^{2s' - m - 2 \\ell}$, we have", "anchor": { "source": "manuscript", "line_start": 2832, "line_end": 2833, "char_start": 167982, "char_end": 168060 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Lemma \\ref{lm:hyperbola-plane} (1) with $A = e^{2s' - m - 2 \\ell}$, we have" }, { "from": "lm:concentrated-surface", "to": "lm:curve-in-N", "type": "proof_dependency", "evidence": "By Lemma \\ref{lm:concentrated-surface}, we have that $\\cC = \\cC(\\cT(y), z)$ is the $\\beta^{-100} e^{-3s'}$-neighborhood of the following curve ($v_1,w_1$ as above):", "anchor": { "source": "manuscript", "line_start": 2884, "line_end": 2885, "char_start": 170579, "char_end": 170743 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Lemma \\ref{lm:concentrated-surface}, we have that $\\cC = \\cC(\\cT(y), z)$ is the $\\beta^{-100} e^{-3s'}$-neighborhood of the following curve ($v_1,w_1$ as above):" }, { "from": "lm:product-w", "to": "lm:curve-in-N", "type": "proof_dependency", "evidence": "By Lemma \\ref{lm:product-w}, we have", "anchor": { "source": "manuscript", "line_start": 1347, "line_end": 1347, "char_start": 69527, "char_end": 69563 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By Lemma \\ref{lm:product-w}, we have" }, { "from": "eq:fvy", "to": "prop:high-dim-decomposition", "type": "proof_dependency", "evidence": "Moreover, we may apply \\eqref{eq:fvy}, \\eqref{eq:fwy}, and \\eqref{eq:fy} to conclude that for", "anchor": { "source": "manuscript", "line_start": 3305, "line_end": 3305, "char_start": 199878, "char_end": 199971 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "Moreover, we may apply \\eqref{eq:fvy}, \\eqref{eq:fwy}, and \\eqref{eq:fy} to conclude that for" }, { "from": "eq:compare-u-orbit", "to": "prop:high-dim-decomposition", "type": "proof_dependency", "evidence": "By \\eqref{eq:compare-u-orbit}, we have that $a(2s')_\\ast\\mu_k$ and $a(2s')_{\\ast}\\bar \\mu_k$ are $O(\\beta)$-close, namely, for any $f \\in C_c^{\\infty} (X)$,", "anchor": { "source": "manuscript", "line_start": 3315, "line_end": 3316, "char_start": 200565, "char_end": 200721 }, "origin": "review_model_inference", "evidence_source_match": true, "evidence_verbatim_match": true, "evidence_match_mode": "exact", "source_evidence": "By \\eqref{eq:compare-u-orbit}, we have that $a(2s')_\\ast\\mu_k$ and $a(2s')_{\\ast}\\bar \\mu_k$ are $O(\\beta)$-close, namely, for any $f \\in C_c^{\\infty} (X)$," } ], "coverage": { "mode": "native_latex_index", "claims": 56, "sections": 12, "external_references": 35, "source_bibliography_entries": 35, "unresolved_bibliography": 0, "unresolved": 0 }, "limitations": [ "Reference edges are not certified proof dependencies. Edge direction is prerequisite \u2192 consumer.", "Numbers reflect simple source counters; labels and line anchors are authoritative.", "Extractive statements and proof openings are marked; they are not reconstructed proof gists.", "Macros, included files and custom counters require an expanded source or annotated fallback." ] } }