A local replacement for Bourgain's periodic Fourier approximation

This concerns equations (4.6)–(4.14) of Jean Bourgain's On the Fourier–Walsh spectrum of the Moebius function, arXiv 1112.1423v1. It is a local repair, not a verification of the whole paper. The original source, independent initial panel assessments, and original dependency graph are preserved.

Sol 6.1 at max effort supplied five explicit replacement lemmas. Opus 5.5 at max independently assessed all five as valid and checked the preservation of the local downstream bounds. Curator inspection also checks the kernel, grid-counting and parameter calculations below. All of these are AI mathematical assessments; no formal certificate or independent human grade is claimed.

What is replaced

Under the periodic Fourier-series interpretation used downstream, the displayed conditions (4.6) force a continuous periodic function to approach opposite values at 0 and 1. They cannot hold uniformly as written for large ℓ. The replacement does not retain that pointwise specification or the exact auxiliary cutoff 2^ℓ. It establishes the required discrete approximation and coefficient bounds, with a larger cutoff that still implies (4.14).

Let h be the original right-continuous periodic step function, equal to +1 on [0,1/2) and −1 on [1/2,1). For an integer ℓ≥1 choose

[ \delta=2^{-\ell-2},\qquad M=2^{2\ell+2},\qquad K_M(s)=\frac1M\left|\sum_{r=0}^{M-1}e^{2\pi i r s}\right|^2, ]

and define the shifted Fejér smoothing

[ p_\ell(t)=\int_{-1/2}^{1/2} K_M(s)h(t+\delta-s),ds. ]

The kernel is nonnegative, has integral 1, and has frequencies |b|<M. Hence p_ℓ is real, periodic and bounded in absolute value by 1. Direct integration gives

[ \widehat h(0)=0,\qquad \widehat h(b)=\frac{1-(-1)^b}{\pi i b}\quad(b\ne0),\qquad \widehat p_\ell(b)=\left(1-\frac{|b|}{M}\right)e^{2\pi i b\delta}\widehat h(b) ]

for |b|<M, and zero otherwise. Thus (\sum_b|\widehat p_\ell(b)|\le(16/\pi)\ell) and (M\le2^{4\ell}).

Why the discrete grid is controlled

The elementary kernel bound (K_M(s)\le1/(4M|s|_{\mathbb T}^2)) gives tail mass at most (1/(2Ma)) outside distance a. Away from both jumps, the approximation error is therefore at most 1/(Ma).

For a dyadic grid of size Q≤1/(2δ), every non-jump point lies at least 2δ from a jump. The shift by +δ also matches h's prescribed value at each jump itself. The pointwise error is at most 1/(Mδ)=2^(-ℓ). For larger grids, the fraction within 2δ of either jump is at most 16δ; the error there is at most 2, and the remaining points have error at most 2^(-ℓ). Consequently, uniformly for dyadic Q≥2,

[ \frac1Q\sum_{k=0}^{Q-1}|h(k/Q)-p_\ell(k/Q)|^2\le17,2^{-\ell}. ]

For A⊂{0,…,m−1}, put (p_A(y)=\prod_{j\in A}p_\ell(y/2^{j+1})), on 0≤y<2^m. Each relevant grid is sampled uniformly. Telescoping the product and using the L² triangle inequality yields

[ 2^{-m/2}|w_A-p_A|_2\le\sqrt{17},|A|,2^{-\ell/2}. ]

The finite expansion has coefficient-product ℓ¹ sum at most ((C_0\ell)^{|A|}), for an absolute C₀≥max(2,16/π). Empty products are exactly 1, with zero error. The replacement uses a non-strict error bound at the empty set; for nonempty sets an enlarged constant restores the printed strict bound. The bit domain 0≤y<2^m follows from the binary coordinates (4.2), correcting the printed m−1 endpoint after (4.3).

Transfer and final frequency bounds

The block coefficient formula is (g_A(z)=2^{-m}\sum_y f(z,y)w_A(y)). Walsh orthogonality and |f|≤1 imply (\sum_A|g_A(z)|^2\le1) and (|g|_2\le\sqrt N), with counting norms. Cauchy–Schwarz then gives

[ |g-\widetilde g|2^2\le17N2^{-\ell}\sum{|A|<K_0}|A|^2. ]

For 1≤m<n/100 and K₀≥1, the final sum is at most ((4/3)n^{K_0}), or zero if no nonempty set is allowed. Thus the replacement even gives an error of order (n^{K_0/2}2^{-\ell/2}\sqrt N), which implies the required (n^{K_0}2^{-\ell/2}\sqrt N) bound.

Taking (\ell=\lceil10K_0\log_2 n\rceil) gives (|b_j|<M\le n^{44K_0}), while the coefficient-product sum is bounded by ((C K_0\log n)^{K_0}). These are the local conclusions of (4.14), with absolute constants. There is no assumption of the failed smoothing claim or an unproved replacement lemma.

This checks the local substitution only. The characterwise Case-I scope of the Balog–Perelli import, the full parameter bookkeeping elsewhere, and completeness of the main theorem remain unresolved in this campaign.

Records and accounting

The complete repair report retains Sol's argument and all five lemma proofs, Opus's individual checks, scope and qualifications. The overlay is separate from the original graph. Human grades remain pending.

Both native calls completed. The checker originally received an invalid-assessment label because its otherwise valid JSON was fenced in Markdown. Offline recovery removed only the complete enclosing fence and retained strict field validation. The original visible answer and private provider bytes are unchanged; previous parsing records are archived. No extra provider or inference call was made.

Successful-call conservative accounting is $0.584531 (Sol $0.176589, Opus $0.407942). Including the earlier rejected delivery's retained $0.508588 allowance, the total repair accounting is $1.093119, below $2. With the initial full-paper reviews, total campaign accounting is $4.664242, below $25. These are conservative token estimates and retained allowances, not invoices.