MathAudit

Paper metadata

Original submission
October 8, 2023
Latest revision
October 9, 2024 · version 2
Claimed MSC
11K31, 11J13
arXiv categories
math.NT
Comments
minor corrections of the previous version

Abstract

This short and simple communication is motivated by recent papers by L. Colzani and A. Kochergin. We give a brief analysis of an example by Poincaré related to sums of the type ∑k=0t−1f(kα+x) \sum_{k=0}^{t-1} f(kα+{x}) where ff is a continuous periodic function and αα is irrationaland its recent generalisations. Most of the constructions under consideration are well-known. In this note, we just wanted to bring all the results together and give a general and improved multi-dimensional formulation of a recent result by A. Kochergin, prove non-existence of a universal continuous function and discuss some of the related results in terms of Diophantine Approximation. In particular, in our opinion smoothness results involving Diophantine exponents ωω, ω^\hatω and λ^\hatλ had never been documented before.

AI-generated audit

Audit summary

Audited against arXiv v2

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections 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 August 20, 2026
01Statements2 reported findingsCorrect

The constructions for Poincare-type and Kronecker sums, including the dense and nowhere-dense alternatives and the Diophantine exponent bound, are correct.

Theorems 1–5Correct

The prescribed behavior of the weighted Kronecker sums is achieved

Pages 1 and 8–18 · Theorems 1–5 · arXiv:2310.05003v2

The nested interval constructions keep earlier partial sums stable while new blocks force the required oscillation or localization. The discrepancy estimate in the final theorem converts the uniform exponent into the stated bound after optimizing the block length.

Full paper, version 2 ↗
Theorem 1 and proof of Theorem 5Typos · no status impact

A coefficient type and one exponent sign are typos

Pages 9 and 15 · constructions for Theorems 1 and 5 · arXiv:2310.05003v2

The coefficients summing to one are positive reals, not positive integers. In Theorem 5 the subsequence supplied by the definition has lower bound ζν≥Mν+1−ω^−δ\zeta_\nu\geq M_{\nu+1}^{-\widehat\omega-\delta}; the printed −ω^+δ-\widehat\omega+\delta reverses the error sign. Using the correct sign and then letting δ\delta tend to zero gives the displayed final exponent.

02Proofs2 reported findingsCorrect

The nested constructions and exponent optimization are complete after the sign correction in the final subsequence bound.

Theorems 4–5Correct and complete after the stated sign repair

The smooth constructions respect the stated Diophantine thresholds

Pages 14–18 · Theorems 4–5 and Section 8 · arXiv:2310.05003v2

For Theorem 4, the Fourier blocks in (40) have the displayed decay, and estimate (43) tends to infinity exactly under d<2ω^−ωd<2\widehat\omega-\omega. For Theorem 5, splitting the Fourier series at Mν+1M_{\nu+1} gives the two terms Q−λ^+δζν−1Q^{-\widehat\lambda+\delta}\zeta_\nu^{-1} and QMν+1n−γQM_{\nu+1}^{n-\gamma}. Using the corrected subsequence bound ζν≥Mν+1−ω^−δ\zeta_\nu\geq M_{\nu+1}^{-\widehat\omega-\delta} and optimizing QQ yields the threshold in (46).

Sections 6–8Correct and complete after the stated repairs

All limiting choices can be made in the announced order

Pages 8–18 · proofs of Theorems 1–5 · arXiv:2310.05003v2

At every stage only finitely many previous constraints must be preserved, so the next denominator and block can be chosen sufficiently large. The corrected exponent sign leaves an arbitrarily small loss that disappears in the final limit.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:2310.05003v2
Authors listed
Nikolay Moshchevitin
Audit date
August 20, 2026
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