Targeted local repair

Target: local-downstream-fourier-bounds. State: assessed.

Statement: established. Supplied proof: defective. Replacement: valid.

These are separate, model-assessed outcomes. Failure to prove leaves the statement unresolved. Conditional imported premises remain conditional. No formal certificate or human grade is inferred.

Blocked inputs and consequences: local-downstream-fourier-bounds, original-discrete-error-consequence, original-fourier-l1-consequence, original-g-error-consequence, original-periodic-smoothing.

Two-wave reservation: $1.440436; attempted-call accounting: $0.584531. Unknown costs retain their reserved allowances. Collection never submits the checker wave.

Local goal and scope

Curator formulation origin: ai_source_adjudication.

A curator-formulated local existence problem preserving (4.8)-(4.14), not a proof of Bourgain's main theorem. The original target excerpt is (4.14); additional exact excerpts specify the remaining local obligations. A replacement may weaken the failed pointwise auxiliary claim and enlarge its cutoff from 2^ell to 2^(C ell), but must establish the original downstream discrete-error and frequency bounds. The bit variable y ranges over 0,...,2^m-1, as follows from the binary decomposition (4.2); the printed m-1 after (4.3) is recorded as a separate domain typo. No additional hypotheses on the main theorem are permitted. The replacement L2 estimate uses <=; for the empty set the error is exactly zero, whereas the first printed strict inequality in (4.9) would read 0<0. For nonempty A an enlarged absolute constant recovers the strict inequality. This corrects a vacuous empty-product endpoint in the auxiliary estimate and does not weaken the final target (4.14).

Construct a replacement for the periodic Fourier approximation, proving the following local conclusions together. These conclusions, not the contradictory auxiliary pointwise specification, are the target. For every positive integer ell, construct a real 1-periodic trigonometric polynomial p_ell with |p_ell|<=1 and integer Fourier frequencies |b|<=2^(C ell), for an absolute C independent of all parameters. Its Fourier coefficients are integral_0^1 p_ell(t) exp(-2 pi i b t) dt and must satisfy sum_b |p_hat_ell(b)|<=C ell. Let h be the supplied periodic step function. For every positive integer m and A subset {0,...,m-1}, set w_A(y)=product_{j in A} h(y/2^(j+1)) and p_A(y)=product_{j in A} p_ell(y/2^(j+1)), for integers 0<=y<2^m. Prove the counting-measure bound 2^(-m/2)||w_A-p_A||_2<=C |A| 2^(-ell/2). This must hold uniformly even for small j whose grids have atoms exactly at both jumps; continuous L2 control alone is insufficient. Empty products equal 1. Prove the finite integer-frequency expansion of p_A and the coefficient-product l1 bound (C ell)^|A|, hence the bound (C ell)^K0 when |A|<K0 and K0>=1. Preserve real-valuedness and the sup norm bound. For n sufficiently large, N=2^n, integer m<n/100, and real K0>=1, let g be the supplied Walsh projection of any f with ||f||_infty<=1, over a block J of m bits, and let g_A(u,w) be its block coefficients as in (4.3). Define tilde g by replacing w_A with p_A. Prove its counting-measure error ||g-tilde g||_2<=C n^K0 2^(-ell/2) sqrt(N). Derive all projection/coefficient norm bounds from the listed elementary inputs; do not use the paper's faulty normalizations as premises. Finally, explain the parameter substitution ell asymptotic to K0 log n (with a fixed sufficiently large constant) and verify the two downstream conclusions of (4.14): each |b_j|<n^(C K0), and the coefficient-product sum bounded by (C K0 log n)^K0. State explicitly the enlarged auxiliary cutoff and justify why every required local conclusion survives with absolute constants. No conclusion about the full main theorem or imported number-theoretic estimates is requested.

Proposed argument

Take h0=p_ell constructed in R1. Lemmas R2–R3 prove the uniform counting-measure approximation, including grids with atoms at both jumps, and the finite Fourier expansion with its coefficient-product bound. Lemmas R4–R5 derive the correctly normalized block coefficient bounds and prove ||g-tilde g||_2 <= C n^K0 2^(-ell/2) sqrt(N).

For the parameter substitution, fix the absolute constant D=10 and choose ell=ceil(D K0 log_2 n). For n>=2 and K0>=1, D K0 log_2 n <= ell <= (D+1)K0 log_2 n, so ell is asymptotic to K0 log n. Every index in the finite expansion satisfies |b_j|<M<=2^(4ell)<=n^(44K0). Put L=C0(D+1)K0 log_2 n, where C0>=2 is the constant from R1. Then L>1, and for |A|<K0 the coefficient-product sum is at most (C0 ell)^|A| <= L^|A| < L^K0. Converting logarithm bases and enlarging one absolute constant C proves both strict inequalities in (4.14). This parameter choice also gives ||g-tilde g||_2 <= C n^(-4K0) sqrt(N).

The enlarged auxiliary cutoff is M-1<2^(2ell+2)<=2^(4ell), rather than 2^ell. Its logarithmic Fourier l1 norm and the required discrete error are proved directly below, so this enlargement changes only absolute constants in the downstream conclusions. Empty products have exactly zero approximation error; thus the requested <= version of (4.9) holds, while a larger constant gives its strict form for nonempty A. The final strict coefficient bound also holds for empty A because its right-hand side exceeds 1.

This proves the explicitly permitted local replacement interpretation: h0 is the constructed polynomial, and (4.14) concerns its finite expansion after the stated parameter choice. No blocked pointwise specification, L-infinity contraction of the Walsh projection, or conclusion about the full main theorem is assumed.

Replacement lemma: R1-shifted-kernel-polynomial

For every integer ell>=1, set delta=2^(-ell-2) and M=2^(2ell+2). There is an explicit real 1-periodic trigonometric polynomial p_ell with |p_ell|<=1, Fourier support |b|<M<=2^(4ell), and sum_b |p_hat_ell(b)|<=C0 ell for an absolute C0>=2. Its coefficients are p_hat_ell(b)=integral_0^1 p_ell(t) exp(-2 pi i b t) dt. Let D_jump={0,1/2}, with distance taken on the circle. Whenever distance(t+delta,D_jump)>=a>0, with a<=1/2, one has |p_ell(t)-h(t+delta)|<=1/(Ma).

Dependencies: periodic-step-definition, elementary-analysis.

Write e(t)=exp(2 pi i t) and define K_M(s)=M^(-1)|sum_{r=0}^{M-1} e(rs)|^2. Expanding the square gives K_M(s)=sum_{|b|<M}(1-|b|/M)e(bs). Consequently K_M is real, nonnegative, 1-periodic, and has integral 1 over any interval of length 1; the integral assertion follows by integrating each integer exponential.

Let ||s||_T denote distance to the nearest integer. The geometric-series identity gives, for nonintegral s, K_M(s)=sin^2(pi M s)/(M sin^2(pi s)). The supplied sine inequalities therefore give K_M(s)<=1/(4M||s||T^2). Hence, for 0<a<=1/2, integral{a<=|s|<=1/2} K_M(s) ds <= 2 integral_a^(1/2) (4Ms^2)^(-1) ds <=1/(2Ma).

Define p_ell(t)=integral_{-1/2}^{1/2} K_M(s)h(t+delta-s) ds. Nonnegativity and unit mass give real-valuedness and |p_ell(t)|<=1. Expanding the finite kernel sum and changing variables in each integral gives p_ell(t)=sum_{|b|<M} c_b e(bt), where c_b=(1-|b|/M)e(b delta)h_hat(b) and h_hat(b)=integral_0^1 h(v)e(-bv)dv. Thus p_ell is a 1-periodic trigonometric polynomial. Integrating this finite expansion against e(-bt) verifies that c_b are precisely the required Fourier coefficients, with coefficients zero outside |b|<M.

Direct integration of the two constant pieces of h gives h_hat(0)=0 and, for b!=0, h_hat(b)=(1-(-1)^b)/(pi i b). Therefore sum_b |c_b| <= (4/pi)sum_{b=1}^{M-1}1/b. Since M=2^(2ell+2), grouping the positive integers into dyadic intervals shows sum_{b=1}^{M-1}1/b <=2ell+2<=4ell. Any fixed C0>=max(2,16/pi) suffices. Also 2ell+2<=4ell, proving the asserted cutoff.

Finally, put v=t+delta. If distance(v,D_jump)>=a, then h(v-s)=h(v) whenever |s|<a; possible equality endpoints do not affect the integral. Since the difference is always at most 2, |p_ell(t)-h(v)| <=2 integral_{a<=|s|<=1/2}K_M(s)ds <=1/(Ma).

Replacement lemma: R2-uniform-dyadic-grid-error

For p_ell from R1 and every power of two Q>=2, Q^(-1)sum_{k=0}^{Q-1}|h(k/Q)-p_ell(k/Q)|^2 <=17·2^(-ell). This includes the prescribed right-sided values of h at 0 and 1/2.

Dependencies: R1-shifted-kernel-polynomial, periodic-step-definition, elementary-analysis.

Use delta and M from R1, so M delta=2^ell. First suppose Q<=(2delta)^(-1). Both jumps are grid points. A grid point t not at a jump has distance at least 1/Q>=2delta from the jump set. Consequently h(t+delta)=h(t) and distance(t+delta,D_jump)>=delta. At either jump itself, the prescribed value of h is its value immediately to the right, so again h(t+delta)=h(t), and the shifted point has distance delta from the jump set. R1 therefore gives |p_ell(t)-h(t)|<=1/(M delta)=2^(-ell) at every grid point. In particular, |p_ell(0)-1| and |p_ell(1/2)+1| are at most 2^(-ell). The mean squared error is at most 2^(-2ell).

Now suppose Q>(2delta)^(-1). Declare a grid point bad when its circular distance from D_jump is at most 2delta. Each of the two corresponding circular arcs has length 4delta. An arc of length L<1 contains at most LQ+2 grid points: after splitting once if it wraps around 0, multiply coordinates by Q and count integers in each ordinary interval. Thus the bad proportion is at most 8delta+4/Q<=16delta. At every other point t, shifting by delta crosses no jump and leaves distance at least delta from the jump set. Hence h(t+delta)=h(t), and R1 again bounds the error by 2^(-ell). At bad points the error is at most 2, because |p_ell|<=1 and |h|=1. The mean squared error is therefore at most 4(16delta)+2^(-2ell) <=17·2^(-ell).

Replacement lemma: R3-products-and-finite-expansion

For every integer m>=1 and A subset {0,...,m-1}, define p_A(y)=product_{j in A}p_ell(y/2^(j+1)) on 0<=y<2^m. Then p_A is real, |p_A|<=1, and 2^(-m/2)||w_A-p_A||2 <=sqrt(17)|A|2^(-ell/2), using counting measure. Writing c_b=p_hat_ell(b), p_A has the finite expansion sum{(b_j): |b_j|<M} (product_{j in A}c_{b_j})e(y sum_{j in A}b_j/2^(j+1)). Its coefficient-product l1 sum is at most (C0 ell)^|A|, hence at most (C0 ell)^K0 whenever |A|<K0 and K0>=1. Empty products are 1.

Dependencies: R1-shifted-kernel-polynomial, R2-uniform-dyadic-grid-error, walsh-product-definition, elementary-analysis.

Fix j<m and put Q=2^(j+1). Since Q divides 2^m, the fractional parts of y/Q, as y runs through 0,...,2^m-1, visit each Q-grid point equally often. R2 therefore gives 2^(-m)sum_y |h(y/Q)-p_ell(y/Q)|^2 <=17·2^(-ell).

Order the elements of A. The difference between the two products is a telescoping sum, each term consisting of one difference h(y/2^(j+1))-p_ell(y/2^(j+1)) multiplied by other h and p_ell factors. Those other factors have absolute value at most 1. The L2 triangle inequality thus gives the asserted normalized product error. For A empty both products are exactly 1, and the error is zero. Reality and the sup norm bound follow directly from the same properties of every factor, and hold for all real y as well.

Substituting the finite expansion from R1 into each factor and multiplying proves the displayed expansion. It is also an integer-frequency expansion for the period 2^m: its term indexed by (b_j) has integer harmonic index k=sum_{j in A}2^(m-j-1)b_j and exponential e(ky/2^m). On the sampled cyclic group, indices may be reduced modulo 2^m. Combining coincident indices cannot increase the l1 norm, by the triangle inequality.

Finite-sum factorization gives sum_{(b_j)}|product_{j in A}c_{b_j}|=(sum_{|b|<M}|c_b|)^|A| <=(C0 ell)^|A|. The empty tuple contributes 1. Since C0 ell>=2, increasing the exponent from |A| to K0 preserves the required upper bound.

Replacement lemma: R4-correct-block-coefficient-norms

Let z=(u,w) denote the n-m outside bits, so there are 2^(n-m) values of z and 2^m values of y. For the supplied projection onto |A intersection J|<K0, every allowed block coefficient satisfies g_A(z)=2^(-m)sum_{y=0}^{2^m-1}f(z,y)w_A(y). Consequently |g_A(z)|<=1, sum_{|A|<K0}|g_A(z)|^2<=1, ||g_A||{2,z}<=2^((n-m)/2), and sum{|A|<K0}||g_A||_{2,z}^2=2^(-m)||g||_2^2<=2^(-m)||f||_2^2<=2^(n-m). In particular, ||g||_2<=||f||_2<=sqrt(N), with all displayed norms using counting measure.

Dependencies: block-projection-definition, block-expansion-definition, binary-block-coordinates, walsh-product-definition, walsh-orthonormality, elementary-analysis.

The binary block coordinates give a bijection between x and (z,y), with y in 0,...,2^m-1. By the definition of h, h(x/2^(j+1)) is the binary sign determined by bit j. Thus a global Walsh character factors as v_T(z)w_A(y), where T indexes outside bits and A indexes the block bits relabeled as 0,...,m-1.

Use the supplied orthonormal Walsh basis to expand f in these tensor-product characters. Its global coefficients are normalized counting inner products, namely 2^(-n)sum_x f(x)w_S(x). The projection retains every outside index T precisely for those block indices A with |A|<K0. For such A, orthogonality in y gives g_A(z)=sum_T f_hat(T union A_J)v_T(z)=2^(-m)sum_y f(z,y)w_A(y), where A_J is the corresponding set of global block positions. This also verifies the coefficient appearing in the supplied block expansion.

The averaging formula and |f|<=1 immediately give |g_A(z)|<=1. For each fixed z, expand f(z,·) in the complete block Walsh basis. Expanding the finite squared norm and applying orthogonality yields sum_{all A}|2^(-m)sum_y f(z,y)w_A(y)|^2=2^(-m)sum_y|f(z,y)|^2<=1. Restricting this sum proves the pointwise selected-coefficient bound.

Likewise, block orthogonality gives ||g||2^2=2^m sum{allowed A}||g_A||_{2,z}^2. Summing the preceding coefficient identity over z proves that this is at most ||f||_2^2<=N. Finally, |g_A(z)|<=1 and the outside domain size 2^(n-m) give the individual coefficient L2 bound. If f is real, its displayed partial coefficients are real.

Replacement lemma: R5-projected-approximation-error

For n sufficiently large, integer 1<=m<n/100, and real K0>=1, define tilde g(z,y)=sum_{A subset {0,...,m-1}, |A|<K0}g_A(z)p_A(y). Then ||g-tilde g||_2 <= C n^K0 2^(-ell/2)sqrt(N), with an absolute C. If f is real, tilde g is real.

Dependencies: R3-products-and-finite-expansion, R4-correct-block-coefficient-norms, elementary-analysis.

Put d_A(y)=w_A(y)-p_A(y). At each (z,y), finite Cauchy-Schwarz and R4 give |sum_A g_A(z)d_A(y)|^2 <= (sum_A|g_A(z)|^2)(sum_A|d_A(y)|^2) <=sum_A|d_A(y)|^2, with sums over allowed A. Summing over z and y, and applying R3, yields ||g-tilde g||2^2 <=2^(n-m)sum_A||d_A||{2,y}^2 <=17N·2^(-ell)sum_A|A|^2.

Let r be the largest integer with 0<=r<=m and r<K0. If r=0, every d_A is zero. Otherwise, the binomial theorem gives k<=2^k, hence k^2<=4^k. Also binom(m,k)<=m^k, directly from its product formula. Therefore sum_A|A|^2=sum_{k=1}^r k^2 binom(m,k) <=sum_{k=1}^r(4m)^k <=(4/3)(4m)^r. Since 4m<n and r<K0, this is at most (4/3)n^K0. The preceding error bound consequently gives even the stronger estimate C n^(K0/2)2^(-ell/2)sqrt(N), which implies the requested bound. All constants are independent of m, n, K0, and ell. Reality when f is real follows from R4 and the real-valued p_A.

Checker evidence

Goal formulation check: it preserves the literal target (4.14), (4.8)-(4.12) and the hypotheses: ell in Z_+, m < n/100, N = 2^n large, ||f||_inf <= 1. It changes only the contradictory auxiliary (4.6)/(4.7). It relaxes the strict sign in (4.9), whose literal empty-set case reads 0 < 0 and is false. Both changes are explicitly supported.

Why the original is defective: a finite-frequency h0 is continuous and 1-periodic. Then (4.6) forces h0(0) >= 1 - C2^-ell and h0(1^-) <= -1 + C2^-ell, which contradict each other for large ell.

The replacement: p_ell is the Fejer smoothing of h shifted by delta = 2^(-ell-2), with M = 2^(2ell+2). Its frequencies satisfy |b| < M <= 2^(4ell), and sum|c_b| <= (16/pi)ell. R1-R5 check line by line:

• R2/R3 give a uniform discrete error bound for products. This includes grids with atoms at both jumps, because the rightward shift matches h's right-sided values.
• R4/R5 give the projection error bound with the correct normalization, without any L-infinity contraction.

Parameter substitution: take ell = ceil(10 K0 log2 n). For n >= 2 and K0 >= 1, this lies in [10K0 log2 n, 11K0 log2 n].

• First part of (4.14): |b_j| < M <= 2^(4ell) <= n^(44K0).
• Second part: with L = 11 C0 K0 log2 n = (11C0/ln 2)K0 ln n > 1, the l1 sum is <= (C0 ell)^|A| <= L^|A| < L^K0. This is (4.14) with C = max(44, 11C0/ln 2). For the empty A the sum is 1 < L^K0.
• In addition, ||g - tilde g|| <= C n^(-4K0) sqrt(N).

No blocked claim and no unproved lemma is used. This is a model assessment, not a certificate.

Check: R1-shifted-kernel-polynomial

Checker assessment: valid.

Kernel facts:

• The identity M^-1|sum_{r<M} e(rs)|^2 = sum_{|b|<M}(1-|b|/M)e(bs) is correct.
• Its closed form is sin^2(pi M s)/(M sin^2(pi s)).
• Since sin(pi||s||) >= 2||s|| (from sin t >= 2t/pi), K_M <= 1/(4M||s||^2).
• The tail mass over a<=|s|<=1/2 is at most (1/(2M))(1/a-2) <= 1/(2Ma).

Fourier coefficients:

• The substitution v = t+delta-s gives c_b = (1-|b|/M)e(b delta)h_hat(b).
• Here h_hat(0) = 0 and h_hat(b) = (1-(-1)^b)/(pi i b).
• Hence sum|c_b| <= (4/pi)H_{M-1} <= (4/pi)(2ell+2) <= (16/pi)ell. The dyadic grouping uses 2ell+2 blocks.
• M = 2^(2ell+2) <= 2^(4ell) holds iff ell >= 1.

Other claims:

• Nonnegativity and unit mass give a real p_ell with |p_ell| <= 1.
• If dist(v,{0,1/2}) >= a, then h(v-s) = h(v) for |s| < a. So |p_ell(t)-h(v)| <= 2*(1/(2Ma)) = 1/(Ma).

Check: R2-uniform-dyadic-grid-error

Checker assessment: valid.

First, M*delta = 2^ell.

Case Q <= 2^(ell+1):

• 0 and 1/2 are grid points.
• Every other grid point is at distance >= 1/Q >= 2delta from them.
• At the jumps, the +delta shift matches the prescribed right-sided values: h(delta) = h(0) = 1 and h(1/2+delta) = h(1/2) = -1, at distance exactly delta.
• So h(t+delta) = h(t) and dist >= delta at every grid point. R1 with a = delta gives pointwise error <= 2^-ell.

Case Q > 2^(ell+1):

• There are at most 2(4delta Q+2) grid points within 2delta of a jump.
• Their fraction is <= 8delta + 4/Q < 16delta = 4*2^-ell, and the squared error there is <= 4.
• Elsewhere the error is <= 2^-ell.
• The mean is <= 162^-ell + 2^-2ell <= 172^-ell.

Check: R3-products-and-finite-expansion

Checker assessment: valid.

Factor bound:

• For j < m, Q = 2^(j+1) divides 2^m, so y mod Q is equidistributed.
• Both functions are 1-periodic, so the 2^m-average equals the R2 grid average.

Product bound:

• Telescoping w_A - p_A gives |A| terms. Each term is one factor difference times factors of modulus <= 1.
• The L2 triangle inequality then gives 2^(-m/2)||w_A-p_A|| <= sqrt(17)|A|2^(-ell/2). The empty A gives exactly 0.

Expansion and l1 bound:

• Multiplying the finite expansions gives (4.10). Each term has integer index sum 2^(m-j-1) b_j.
• The l1 sum factorizes: sum|prod c_{b_j}| = (sum|c_b|)^|A| <= (C0 ell)^|A|.
• Since C0 ell >= 2 > 1, this is < (C0 ell)^K0 for |A| < K0.
• Reality and |p_A| <= 1 are inherited from the factors.

Check: R4-correct-block-coefficient-norms

Checker assessment: valid.

Coefficient formula:

• h(x/2^(j+1)) = (-1)^(x_j), so via (4.2) each global character factors as v_T(z)w_A(y).
• The projection condition depends only on A, so all T are kept for allowed A.
• Expanding f in the orthonormal basis and using orthogonality in y gives g_A(z) = sum_T f_hat(T u A_J)v_T(z) = 2^-m sum_y f(z,y)w_A(y).
• This is unique by linear independence of the w_A, so it matches the coefficients in (4.3).

Norm bounds:

• |g_A| <= 1.
• Bessel's inequality for fixed z gives sum_{|A|<K0}|g_A(z)|^2 <= 2^-m sum_y|f|^2 <= 1.
• Orthogonality in y gives ||g||^2 = 2^m sum_A||g_A||^2 <= ||f||^2 <= N.
• No L-infinity contraction of the projection is used.

Check: R5-projected-approximation-error

Checker assessment: valid.

Main estimate:

• Pointwise Cauchy-Schwarz with sum_A|g_A(z)|^2 <= 1 gives ||g - tilde g||^2 <= 2^(n-m) sum_A||d_A||^2.
• By R3 this is <= 17 N 2^-ell sum_{|A|<K0}|A|^2.

Counting allowed sets:

• Let r < K0 be the largest allowed size, with r <= m.
• Using k^2 <= 4^k and binom(m,k) <= m^k, sum k^2 binom(m,k) <= sum_{k<=r}(4m)^k <= (4/3)(4m)^r.
• Since 4m < n and r < K0, this is < (4/3)n^K0. If r = 0, all d_A vanish.

Conclusion:

• ||g - tilde g||_2 <= sqrt(68/3) n^(K0/2) 2^(-ell/2) sqrt(N) <= C n^K0 2^(-ell/2) sqrt(N).
• Reality follows from R4 and the real-valued p_A.

Checker qualifications

• Non-blocking: the literal auxiliary specifications (4.6) and (4.7) are not met, and (4.6) cannot be met. The candidate has frequency cutoff M-1 = 2^(2ell+2)-1 <= 2^(4ell), not 2^ell. Its pointwise control holds only away from the jumps. The goal formulation explicitly permits both changes, and they affect only the absolute constants in the target (4.14).
• Non-blocking: R5 is stated for 1 <= m. The case m = 0 is trivial: only A = empty set occurs, so tilde g = g. It is also excluded by log B < m whenever B >= 1.
• Non-blocking: the second inequality in (4.9), C|A|2^(-ell/2) < C K0 2^(-ell/2), is not restated. It is immediate from |A| < K0, and the empty-set case reads 0 <= 0 < C K0 2^(-ell/2).
• Cosmetic: the main argument says the cutoff consequences are proved 'directly below', but they are proved in the separately listed lemmas R1-R5.