MathAudit

Paper metadata

Original submission
October 4, 2008
Claimed MSC
arXiv categories
math.NT
Comments
5 pages

Abstract

By means of Peres-Schlag's method we prove the existence of real numbers α,βα,β such that lim inf⁡q→∞q(log⁡q)2∥αq∥ ∥βq∥>0.\liminf_{q\to\infty}q(\log q)^2\lVert αq\rVert\,\lVert βq\rVert>0.

AI-generated audit

Audit summary

Audited against arXiv v1

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
01Statements1 reported findingCorrect

The paper correctly constructs real numbers whose Littlewood product, after the stated logarithmic factor, has positive lower limit.

Theorem 1Correct

Logarithmically weighted Littlewood lower bound

Pages 1–5 · Theorem 1 and Sections 2–4 · arXiv:0810.0777v1

The lattice-point estimate gives the required uniform sum bound for a fixed badly approximable first coordinate. The dyadic dangerous-set construction and its inductive measure estimate leave a nonempty compact intersection, and every point in that intersection satisfies the claimed lower bound for all sufficiently large denominators.

02Proofs3 reported findingsCorrect

The lattice count and nested-set induction verify the theorem. Two malformed superscripts/subscripts are uniquely determined typographical errors.

Lemma 1Correct

Uniform reciprocal-sum estimate

Pages 1–3 · Lemma 1 and Equations (1)–(3) · arXiv:0810.0777v1

The dyadic decomposition separates the denominator size from ∥αx∥\lVert\alpha x\rVert. Bad approximability gives the required separation of primitive lattice points, while the one-dimensional and polygon cases bound the number of integer points in every cell. Summing those bounds gives the displayed logarithmic estimate uniformly in q<Qq<Q.

Nested dangerous-set constructionCorrect

The surviving compact intersection enforces the lower bound

Pages 3–5 · Sections 3–4 and Lemma 2 · arXiv:0810.0777v1

Each dangerous interval is covered at its assigned dyadic scale, and Lemma 1 bounds the proportion removed between consecutive denominator blocks. Choosing the initial denominator sufficiently large leaves positive measure at every stage. The retained closed sets are nested, so their intersection is nonempty; every point in it avoids all large-denominator dangerous intervals and satisfies the theorem.

Lemma 1 notationTypo

The inverse of the Diophantine constant has a missing exponent marker

Page 3 · end of the proof of Lemma 1 · arXiv:0810.0777v1

Replace each printed δ{−1}\delta\{-1\} by δ−1\delta^{-1}. The same factor is stated correctly in Lemma 1 and follows directly from the preceding lattice-point bound.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:0810.0777v1
Authors listed
Nikolay Moshchevitin
Audit date
August 20, 2026
  1. 01Establish identityMatch an authenticated scholarly identity to this paper.
  2. 02Submit evidenceIdentify the finding and give a formal mathematical response.
  3. 03Re-evaluateA separate agent checks the response and records a disposition.
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