MathAudit

Paper metadata

Original submission
April 24, 2010
Claimed MSC
11J13
arXiv categories
math.NT
Comments
21 pages

Abstract

We give a simplified exposition of the easiest case of a breakthrough result by D.Badziahin, A.Pollington and S.Velani related to W.M.Schmidt's conjecture.

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 explicit vertical-line instance of the Badziahin–Pollington–Velani construction is correct.

Proposition 1Correct

Uniform avoidance of all rational-line intervals

Page 2 · Proposition 1 · arXiv:1004.4269v1

Under inf⁡q≥1q2∥qθ∥≥δ\inf_{q\geq1}q^2\lVert q\theta\rVert\geq\delta and 0<δ≤2−16220<\delta\leq2^{-1622}, the nested construction produces ξ\xi satisfying ∥Aθ−Bξ∥max⁡(A2,B2)≥δ\lVert A\theta-B\xi\rVert\max(A^2,B^2)\geq\delta for every nonzero integer pair (A,B)(A,B). The B=0B=0 case is exactly the hypothesis, while B≠0B\neq0 is covered by the forbidden intervals.

02Proofs3 reported findingsCorrect

The concurrency lemmas, deletion counts, and final nested-interval estimate correctly prove Proposition 1.

Sections 7–8Correct and complete

Bounded-slope line families

Pages 5–14 · Lemmas 1–9 and Fundamental Lemma 1 · arXiv:1004.4269v1

The determinant argument forces all admissible lines meeting one construction interval through a single rational point. The Diophantine lower bound on θ\theta then supplies the required denominator estimate, and the resulting cluster length plus the single exceptional line deletes at most the stated O(R52/55log⁡R)O(R^{52/55}\log R) children.

Section 9Correct and complete

Large-slope decomposition

Pages 14–20 · modified lemmas and Fundamental Lemma 2 · arXiv:1004.4269v1

The parameter ll partitions every remaining height regime. Lemma 3∗3^* again gives concurrency, the lower bound for the rational denominator controls the common cluster, and the principal family, exceptional line, and lines whose forbidden intervals barely reach the parent interval together remove at most 8R52/558R^{52/55} children.

Section 10Correct and complete

Positive branching and nonempty intersection

Pages 20–21 · proof of Proposition 1 · arXiv:1004.4269v1

Induction with R≥2422R\geq2^{422} bounds the delayed losses by a convergent geometric tail and yields Tn+1≥Tn(R−214R52/55log⁡R)>0T_{n+1}\geq T_n(R-2^{14}R^{52/55}\log R)>0. The finite unions are nested compact sets, so their intersection is nonempty. With R=2422R=2^{422}, the assumed δ\delta is strictly below the auxiliary bound required in Section 4.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:1004.4269v1
Authors listed
Nikolay G. Moshchevitin
Audit date
August 20, 2026
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