MathAudit

Paper metadata

Original submission
October 12, 2026
Latest revision
October 21, 2026 · version 2
Claimed MSC
05D10, 37B20, 11J54, 11B25
arXiv categories
math.CO, math.DS, math.NT
Paper license
CC BY 4.0 ↗
Comments
17 pages and 1 figure

Abstract

A set SS of positive integers is 22-large if every two-coloring contains arbitrarily long monochromatic arithmetic progressions with common difference in SS. We prove that every such set meets every nil-Bohr neighborhood of 00. This proves a conjecture of the first author and gives a sharp answer to a question of Host, Kra, and Maass. We also show that, for every fixed nil-Bohr neighborhood BB of 00 and every ε>0\varepsilon>0, restricting the common difference to BB increases the extremal number for kk-term arithmetic progressions by at most a factor of 1+ε1+\varepsilon, provided kk is sufficiently large.

AI-generated audit

Audit summary

Audited against arXiv v2

Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated October 23, 2026
01Statements2 reported findingsNo statement issues found

The shell-coloring theorem and its nil-Bohr recurrence and density consequences are supported for the stated weighted polynomial configurations.

Theorem 1.9No issue found

Every two-coloring produces the required monochromatic weighted configuration

PDF pp. 3–4 and 8–15 · Theorem 1.9 · arXiv:2610.16605v2

The weighted polynomial matrix class is stable under the prescribed operations. Compactness of the normalized curve family and the interpolation lemma produce a uniformly controlled parameter, while the leading-coefficient and last-column lemmas decode the resulting matrix relation into exactly the configuration stated in the theorem.

Theorems 1.7 and 1.8; Corollary 1.10No issue found

Nil-Bohr recurrence and density follow from the coloring theorem

PDF pp. 2–5 and 15–16 · arXiv:2610.16605v2

Coloring successive shells in a nilmanifold converts Theorem 1.9 into a return-time statement for every neighborhood of the identity. The Følner averaging argument then yields the asserted positive-density recurrence, and the displayed specialization gives the corollary.

02Proofs2 reported findingsNo proof issues found

The rounding, compactness, interpolation, coefficient-control, and decoding arguments supporting the central coloring theorem are correct and complete.

Lemmas 3.2 and 3.6No issue found in the reviewed proof

Rounding preserves the weighted filtration and interpolation is uniform

PDF pp. 8–12 · arXiv:2610.16605v2

The rounding error lands in strictly higher weighted degree, so it cannot change the leading data used later. Compactness of the normalized curve family makes the interpolation constants uniform and supplies the finite choice needed by the shell argument.

Proposition 3.7 and Lemma 3.8No issue found in the reviewed proof

The coefficient and last-column calculations close the construction

PDF pp. 12–15 · arXiv:2610.16605v2

The induction isolates the leading weighted coefficients with the asserted nonvanishing and divisibility properties. Reading the last matrix column then gives the desired shifts with all indices and parameter ranges matching Theorem 1.9.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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Paper
arXiv:2610.16605v2
Authors listed
Ryan Alweiss, Sohail Farhangi
Audit date
October 23, 2026
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