# Reading roadmap: Generalization of Selberg's 3/16 Theorem and Affine Sieve

The paper passes from finite congruence expansion to a uniform spectral gap for hyperbolic surfaces with critical exponent greater than one half, then to uniform lattice counts. For smaller critical exponent, symbolic dynamics, convolution flattening and congruence transfer operators supply resonance-free regions and lattice counts. A combinatorial sieve uses the counts to obtain prime-value upper bounds and almost-prime lower bounds in matrix orbits.

Source SHA-256: `430f4bd4a754824f0521bf447c411e4e8b64fac315547a4e8a870adff3df9afc`. Indexed 86,667 characters.

Edges point from prerequisite to consumer. Solid proof edges are attributed inferences, not formal certificates; references remain candidates.

## Statements and separate branches

Read spectral transfer, both counting regimes and the two sieve conclusions with their exact quantifiers.

Claims: t4, t0, t4a, t5a, t5b, t2, t22

## Large critical exponent

Inspect expansion-to-spectrum transfer and the use of Lax–Phillips estimates for congruence counting.

Claims: t4, t0, t4a

## Small critical exponent

Trace symbolic coding, convolution flattening, uniform transfer-operator bounds and contour shifts to the resonance and counting statements.

Claims: t5a, t5b, l1, l1q, props3, p1, p2, p3, p4, pt1

## Sieve applications

Check local densities, levels of distribution, polynomial factorization conventions and the two sieve bounds.

Claims: t:cs, t2, t22

## Internal claim index

### Theorem 1.1 (Acta; native t4) — `t4`

Source lines 347–357.

Let $\Lambda$ be a finitely generated subgroup of $\SL(2, \zed)$ with $\delta(\Lambda) > \frac{1}{2}$. For $q \ge 1$ let $\Lambda(q)$ be the ``congruence'' subgroup $\{x \in \Lambda : x \equiv I \mod q\}$. There is $\varepsilon = \varepsilon(\Lambda) > 0$ such that $$\lambda_1(\Lambda(q)) \ge \lambda_0(\Lambda(q)) +\varepsilon, $$ for all square-free $q \ge 1$ (note that $\lambda_0(\Lambda(q))=\lambda_0(\Lambda)$).

*Extractive statement; proof gist not independently reconstructed.*

### Theorem 1.2 (Acta; native t0) — `t0`

Source lines 370–383.

Let $\Lambda = \langle S \rangle$ be a finitely generated subgroup of $\SL(2, \reals)$ with $\delta(\Lambda) > \frac{1}{2}$. Let $\{N_i\}$ be a family of finite index normal subgroups of $\Lambda$. Then the following are equivalent \item The Cayley graphs $\gra(\Lambda/ N_i, S)$ form a family of expanders. \item There is $\varepsilon = \varepsilon(\Lambda) > 0$ such that $\lambda_1(\Lambda/ N_i) \ge \lambda_0(\Lambda/ N_i) +\varepsilon $.

*Extractive statement; proof gist not independently reconstructed.*

### Theorem 1.3 (Acta; native t4a) — `t4a`

Source lines 400–412.

Let $\Lambda$ be a finitely generated subgroup of $\SL(2, \zed)$ with $\delta(\Lambda) > \frac{1}{2}$. Assume that $q$ is square free and $(q, q_0) =1$, where $q_0$ is provided by the strong approximation theorem \cite{mvw}. There is $\varepsilon_1 > 0$ depending on $\Lambda$ such that for any $g \in \SL_2(q)$ we have \beq &|\{\gamma \in \Lambda \, | \, \|\gamma\| \leq T \, \text{and} \, \gamma \equiv g \, \nod \, q \} | \\ & = \frac{ c_{\Lambda} T^{ 2 \delta}}{|\SL_2(q)|} + O (q^3 T^{2 \delta - \varepsilon_1}). \eeq

*Extractive statement; proof gist not independently reconstructed.*

- `t4` (proof_dependency; curated_source_check): Combining Theorem \ref{t4} with

Candidate references: `external:mvw` (external_reference).

### Theorem 1.4 (Acta; native t5a) — `t5a`

Source lines 432–443.

Let $\Lambda$ be a finitely generated subgroup of $\SL(2, \zed)$ with $\delta(\Lambda) \leq \frac{1}{2}$. For $q \ge 1$ square free let $\Lambda(q)$ be the ``congruence'' subgroup $\{x \in \Lambda : x \equiv I \mod q\}$; let $X(q) = \Lambda(q) \bs \hyp$. There is $\varepsilon = \varepsilon(\Lambda) > 0$ such that $R_{X(q)}(s)$ is holomorphic( with the exception of simple pole at $s= \delta$) for $\Re(s) > \delta - \varepsilon \min \left(1, \frac{1}{\log(1 + |\Im s|)}\right)$.

*Extractive statement; proof gist not independently reconstructed.*

- `p1` (proof_dependency; curated_source_check): applying theorem
\ref{p1} to $E_{q_1}$ for all $q_1 |q$
- `p3` (proof_dependency; curated_source_check): and Proposition \ref{p3}
to the constant function

### Theorem 1.5 (Acta; native t5b) — `t5b`

Source lines 476–488.

Let $\Lambda$ be a finitely generated subgroup of $\SL(2, \zed)$ with $0< \delta(\Lambda) \leq \frac{1}{2}$. Assume that $q$ is square free and $(q, q_0) =1$, where $q_0$ is provided by the strong approximation theorem \cite{mvw}. There is $\varepsilon_1 > 0$, $C > 0$ depending on $\Lambda$ such that for any $g \in \SL_2(q)$ we have \beq &|\{\gamma \in \Lambda \, | \, \|\gamma\| \leq T \, \text{and} \, \gamma \equiv g \, \nod \, q \} | \\ & = \frac{ c_{\Lambda} T^{ 2 \delta}}{|\SL_2(q)|}\left(1 + O\left(T^{-\frac{1}{\log \log T}}\right)\right) 

*Extractive statement; proof gist not independently reconstructed.*

- `pt1` (proof_dependency; curated_source_check): result, of which Theorem \ref{t5b} is an immediate consequence.

Candidate references: `external:mvw` (external_reference).

### Theorem 1.6 (Acta; native t2) — `t2`

Source lines 528–535.

Let $\Lambda$ be a subgroup of $\SL(2, \zed)$ which is Zariski dense in $\SL_2$ and let $f \in \zed[x_{ij}]$ be weakly primitive with $t(f)$ irreducible factors. Then \beq \pi_{\Lambda, f}(T) \ll \frac{N_{\Lambda}(T)}{(\log T)^{t(f)}}. \eeq

*Extractive statement; proof gist not independently reconstructed.*

- `t5b` (proof_dependency; curated_source_check): Theorem \ref{t5b}  we then obtain
- `ontr` (proof_dependency; curated_source_check): Finally, to verify the third axiom concerning the sieve dimension,
we have, using \eqref{ontr}, that

### Theorem 1.7 (Acta; native t22) — `t22`

Source lines 543–553.

Let $\Lambda$ be a subgroup of $\SL(2, \zed)$ which is Zariski dense in $\SL_2$ and let $f \in \zed[x_{ij}]$ be weakly primitive with $t(f)$ irreducible factors. Then there is an $r<\infty$, which can be given explicitly in terms of $\varepsilon(\Lambda)$ in theorems \ref{t4} and \ref{t5a}, such that \beq |\{x \in \Lambda; |x| \le T, \, \, \text{and} \, \, f (x) \, \text{has at most $r$ prime factors} \}| \gg \frac{N_{\Lambda}(T)}{(\log T)^{t(f)}}. \eeq

*Extractive statement; proof gist not independently reconstructed.*


Candidate references: `t4` (internal_reference); `t5a` (internal_reference).

### Theorem [tApol] — `tApol`

Source lines 802–809.

Let $\Lambda$ be a geometrically-finite subgroup of $\SL_2(\zed[\sqrt{-1}])$ with $\delta(\Lambda)>1$ and such that the traces of elements of $\Lambda$ generate the field $\ratls(\sqrt{-1})$. There is $\varepsilon = \varepsilon(\Lambda)> 0$ such that $$\lambda_1(\Lambda(\mathcal{A})) \ge \lambda_0(\Lambda(\mathcal{A})) +\varepsilon$$ as $\mathcal{A}$ varies over squarefree ideals in $\zed[\sqrt{-1}]$.

*Extractive statement; proof gist not independently reconstructed.*

### Theorem [tr] — `tr`

Source lines 947–968.

Let $f \in \mF_{\rho}$ be a real-valued function. \item There is a simple eigenvalue $\lambda_{f} > 0$ of $\mL_{f}: \mF_{\rho} \to \mF_{\rho}$ with strictly positive eigenfunction $h_{f}$. \item The rest of the spectrum of $\mL_{f}$ is contained in $\{z \in \cx \, :\, |z| \leq \lambda_{f} - \varepsilon \}$ for some $\varepsilon > 0$. \item There is a Borel probability measure $\nu_{f}$ on $\Sigma$ such that $\mL_f^{*} \nu_f = \lambda_{f} \nu_f$. \item If $h_f$ is normalized so that $\int h_f d \nu_f =1$ then for every $g \in C(\Sigma)$ $$\lim_{

*Extractive statement; proof gist not independently reconstructed.*

### Theorem [ln] — `ln`

Source lines 1067–1084.

\item There is $\varepsilon > 0$ such that for $\Re z < - \delta + \varepsilon$, $ z \notin U$ ($U$ a suitable neighborhood of $- \delta$), we have \beq \|\mL_z|_{\mathcal{F}_{\rho}(\Sigma)}\|_{\rho} \lesssim |\Im z|^2 e^{-\varepsilon n}. \eeq \item For $ z \in U$ decompose on $\mathcal{F}_{\rho}(\Sigma)$ \beq \mL_z =\lambda_z(\nu_z \otimes h_z) + \mL_{z}'', \eeq (where $z \to \lambda_z, z \to h_z, z \to \nu_z$ are holomorphic extensions to $U$, satisfying $$\mL_z h_z= \lambda_z h_z, \quad \mL^{*}_z \nu_z = \lambda_z \nu_z, \quad \int h_z d \nu

*Extractive statement; proof gist not independently reconstructed.*

### Lemma [l1] — `l1`

Source lines 1249–1258.

Given $\kappa>0$, there is $\kappa'>0$ such that if $\mu \in \mathcal{M}_{\cx}(G)$ satisfies \beq \|\mu\|_{\infty} < p^{-\kappa} \|\mu\|_{1} \eeq and \beq |\mu|(a G_1) < p^{-\kappa} \|\mu\|_1 \eeq for cosets of proper subgroups $G_1$ of $G$, then \beq \|\mu \ast \varphi\|_2 \leq C p^{-\kappa'} \|\mu\|_{1} \|\varphi\|_2 \quad \text{for} \quad \varphi \in L_0^2(G). \eeq Here $p$ is assumed to be sufficiently large.

*Extractive statement; proof gist not independently reconstructed.*

### Lemma [l1q] — `l1q`

Source lines 1273–1283.

Let $q$ be square free and $G=\SL_2(q)$. For $\mu \in \mathcal{M}_{\cx}(G)$ and $q_1|q$ define $|||\pi_{q_1}(\mu)|||_{\infty}$ to be the maximum weight of $|\mu|$ over cosets of subgroups of $\SL_2(q_1)$ that have proper projection in each divisor of $q_1$. Given $\kappa>0$, there is $\kappa'>0$ such that if $\mu$ satisfies for all $q_1 | q$ \beq |||\pi_{q_1}(\mu)|||_{\infty} < q_1^{-\kappa} \|\mu\|_{1} \eeq then \beq \|\mu \ast \varphi\|_2 \leq C q^{-\kappa'} \|\mu\|_{1} \|\varphi\|_2 \quad \text{for} \quad \varphi \in E_q. \eeq

*Extractive statement; proof gist not independently reconstructed.*

### Lemma [props3] — `props3`

Source lines 1517–1525.

Notation being as above, there is $\varepsilon > 0$ such that \beq \| \mM_z^m |_{\mathcal{F}'_{\rho}} \|_{\rho} < q^C e^{-\varepsilon m} \lambda_{\Re z}^m \, \, \text{if} \, \, |\Im z| \leq q \eeq and \beq \| \mM_z^m |_{\mathcal{F}'_{\rho}} \|_{\rho} < |\Im z|^C e^{-\varepsilon \frac{\log q}{\log | \Im z|} m} \lambda_{\Re z}^m \, \, \text{if} \, \, |\Im z| > q. \eeq

*Extractive statement; proof gist not independently reconstructed.*

- `l1q` (proof_dependency; curated_source_check): Applying Lemma \ref{l1q}, it follows that
- `tr` (proof_dependency; curated_source_check): By Ruelle's Theorem   (Thm. \ref{tr}),

### Theorem [p1] — `p1`

Source lines 1564–1573.

The resolvent $(I- \mM_z)^{-1}|_{\mF'_{\rho}}$ is holomorphic on the complex region $D(q)$ given by \beq \Re z < - \delta + \vge_2 \min\left(1, \frac{\log q}{\log(|\Im z| +1)} \right) \eeq (with $\vge_2$ independent of $q$) and satisfies the estimate \beq \| (I- \mM_z)^{-1}|_{\mF'_{\rho}}\| <(q+ | \Im z |)^C. \eeq

*Extractive statement; proof gist not independently reconstructed.*

- `props3` (proof_dependency; curated_source_check): We now use Lemma \ref{props3}  to estimate the resolvent

### Theorem 13.1 (Acta; native t:cs) — `t:cs`

Source lines 1915–1922.

Assume ($A_0$), ($A_1$) and ($A_2$) for $s > 9 t$ and $z=D^{1/s}$ and $X$ large we have \beq \frac{X}{(\log X)^t} \ll S(A, P_z) \ll \frac{X}{(\log X)^t}. \eeq

*Extractive statement; proof gist not independently reconstructed.*

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## Coverage and limits

{"mode": "native_latex_index", "claims": 19, "sections": 13, "external_references": 37, "source_bibliography_entries": 37, "unresolved_bibliography": 0, "unresolved": 0, "numbering": "Native source labels; no inferred printed theorem numbers."}

- Reference edges are not certified proof dependencies. Edge direction is prerequisite → 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.
- Publication adds only selected source-supported proof-input edges. The reference graph and proof graph are incomplete; terminal status is relative to recorded edges. Native IDs remain authoritative; Acta correspondences are separately labeled.