MathAudit

Paper metadata

Original submission
April 11, 2012
Claimed MSC
11J20
arXiv categories
math.NT
Comments
5 pages

Abstract

We prove a conjecture due to Stephen Harrap on inhomogeneous linear Diophantine approximation related to BAD(α,β){\rm BAD}(α,β) sets.

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
01Statements2 reported findingsCorrect

The nonemptiness and full-Hausdorff-dimension conclusions for the weighted twisted set are correct; the definition contains a harmless repeated coordinate.

Main theoremCorrect

The weighted twisted set has full dimension

Pages 1-5 · theorem and Sections 2-3 · arXiv:1204.2561v1

The best-approximation sequence splits into two families with lacunarity after at most 28 terms. At each anisotropic subdivision, only O(R2log⁡R)O(R^2\log R) of the R3R^3 children meet a new resonance strip, leaving a positive branching Cantor set. The transference bound (1) sends its uniform dual separation to BADΘ(2/3,1/3)\mathrm{BAD}_\Theta(2/3,1/3), and the branching estimate tends to ambient dimension two as R→∞R\to\infty.

Full paper, version 1 ↗
Definition of $\mathrm{BAD}_\Theta$Typo · no status impact

The second inhomogeneous coordinate is a typo

Page 1 · displayed definition of BADΘ(α,β)\mathrm{BAD}_\Theta(\alpha,\beta) · arXiv:1204.2561v1

The second term is printed ∥qθ2−η1∥\|q\theta_2-\eta_1\|; it must be ∥qθ2−η2∥\|q\theta_2-\eta_2\|. The vector notation, every later dual form, and the two-coordinate conclusion uniquely determine this correction.

02Proofs2 reported findingsCorrect

The lacunarity count, resonance-strip estimate, nested construction, and transference step are correct and complete.

Equations (3)-(6)Correct and complete

The best-approximation families are uniformly lacunary

Pages 2-3 · equations (3)-(6) · arXiv:1204.2561v1

Partitioning the annular difference of two weighted approximation bodies into 2×282\times28 translates and using central symmetry forces Mν+28≥2MνM_{\nu+28}\geq2M_\nu. Each of the two coordinate-dominant subsequences inherits the same bound, which limits the number of resonances in one scale block by O(log⁡R)O(\log R).

Section 3Correct and complete

The dangerous-child counts leave a nonempty full-dimension construction

Pages 3-5 · equations (9)-(13) and final count · arXiv:1204.2561v1

For the first family, fixing the second coordinate confines every dangerous child to a horizontal interval of controlled length; the second family is symmetric with the coordinates exchanged. The resulting bounds 5R25R^2 and 6R26R^2 per resonance, combined with lacunarity, are smaller than the R3R^3 available children for large RR.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

Detailed audit reportFull reasoning, manuscript locations, and references.
Open report PDF ↗

Author response

Challenge an audit finding

Local workflow preview

A listed author may submit formal evidence that an audit is inaccurate. The response would be considered in a fresh AI re-evaluation; it would not edit the audit automatically.

Paper
arXiv:1204.2561v1
Authors listed
Nikolay G. 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.
Recommended production method

Authenticate with ORCID, then require an exact arXiv match

MathAudit should accept the identity only when ORCID OAuth authenticates the claimant's iD and this exact arXiv paper appears in arXiv's public authority feed for that iD. A matching name alone is not sufficient.

ORCID OAuth and arXiv authority-record lookup are not connected in this local prototype.

Email fallback for papers without a linked ORCID

A production fallback could send a one-time link only when the submitted address matches an independently maintained author-contact allowlist for this paper. MathAudit must return the same message for every address so the form cannot reveal which contacts are on that list.

This demonstration does not send, store, or compare the address.

Structured response preview

This form remains unavailable until production identity verification succeeds. Nothing entered here is submitted.

This panel never establishes authorship in the local prototype. A production result should be described narrowly as an authenticated ORCID match or control of a separately allowlisted author-contact mailbox.