MathAudit

Paper metadata

Original submission
April 17, 2023
Claimed MSC
11J13
arXiv categories
math.NT
Comments
Submitted to Colloquium Mathematicum

Abstract

We give an easy optimal bound for the dimension of the subspaces generated by the best Diophantine approximations.

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 paper gives the correct optimal eventual-span dimensions for best approximation vectors of completely irrational good matrices.

Theorem 1Correct

All four parameter regimes have the stated minimal eventual dimension

Pages 3–8 · Theorem 1 and Sections 4–6 · arXiv:2304.08601v1

Complete irrationality supplies the lower-dimensional obstruction, while the explicit embeddings built from rapidly decreasing irrationality functions make the best-approximation vectors eventually lie in subspaces of exactly the asserted dimensions. The two-column cases use the appropriate Jarnik existence input.

Full paper, version 1 ↗
Setup and cross-referencesTypos · no status impact

The matrix dimensions and several theorem labels are typos

Pages 1, 4, 6, and 8 · setup and constructions for (3b)–(4b) · arXiv:2304.08601v1

The opening display gives xx exactly mm coordinates, so it belongs to Rm\mathbb R^m, and the displayed array for Θ\Theta is n×mn\times m, not m×nm\times n. Several later references to Lemma 5 must be to Lemma 3, the lemma that actually supplies the extension; the Section 5 reference to Theorem 3 must be to Theorem 1. Finally, the two-column construction in Section 6 must invoke part (a), not part (b), of the quoted Jarnik theorem. Each repair is fixed by the referenced hypotheses and surrounding dimensions.

02Proofs2 reported findingsCorrect

The lower-bound geometry and sharpness constructions are complete after the local dimension and citation-label corrections.

Lemmas 3–4Correct and complete

The metric extensions preserve complete irrationality

Pages 5–9 · Lemmas 3–4 and the constructions in Sections 5–6 · arXiv:2304.08601v1

The convergence conditions make the competitor events summable, so the Borel–Cantelli argument leaves a full-measure set of added rows or columns for which the prescribed vectors remain the best approximations. Removing the countable rational-dependence hyperplanes then gives a completely irrational extension with the claimed eventual span.

Sections 4–6Correct and complete after the stated repairs

The lower bounds and realizing examples match

Pages 4–9 · proof of Theorem 1 · arXiv:2304.08601v1

The rational-subspace intersection argument rules out any smaller eventual span. For the upper bounds, the selected irrationality functions separate the intended best approximations from all competitors and preserve complete irrationality, so the constructed span dimensions are attained.

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:2304.08601v1
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.
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.