MathAudit

Paper metadata

Original submission
November 24, 2021
Claimed MSC
11J13
arXiv categories
math.NT
Comments
22 pages

Abstract

We prove a result related to Dirichlet spectrum for simultaneous approximation to two real numbers in Euclidean norm and badly or very well approximability.

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
01Statements3 reported findingsCorrect

The Dirichlet-spectrum realization by vectors with arbitrarily large best-denominator ratios and the approximation by badly approximable vectors are correct.

Theorem 3Correct

The general inductive cylinders realize the prescribed interval data

Pages 6–22 · Theorem 3 and Sections 6–11 · arXiv:2111.12259v1

The recursive choice of primitive lattice points keeps the normalized cylinder volumes inside the prescribed shrinking intervals, while the empty-cylinder and determinant conditions identify the complete best-approximation sequence. The bounds on the auxiliary parameters give the asserted control of consecutive heights at every stage.

Full paper, version 1 ↗
Theorem 1Correct

Every Dirichlet-spectrum value is realized with arbitrarily large denominator ratios

Pages 4 and 7 · Theorem 1 and its deduction from Theorem 3 · arXiv:2111.12259v1

Choosing shrinking parameter intervals with limit λ\lambda fixes the limiting normalized volume at the prescribed spectrum value. The free integer parameters kνk_\nu can simultaneously be taken so large that qν+1/qνq_{\nu+1}/q_\nu dominates the arbitrary function φ(ν)\varphi(\nu). Thus the constructed vector has the stated Dirichlet value and is not badly approximable.

Theorem 2Correct

The bounded-ratio realization has the stated quadratic dependence

Pages 5 and 7 · Theorem 2 and its deduction from Theorem 3 · arXiv:2111.12259v1

With constant interval width proportional to ε\varepsilon and the explicit admissible choice of kk, Theorem 3 places the Dirichlet value between λ−ε\lambda-\varepsilon and λ\lambda. The displayed parameter bounds give m∣⋅∣(θ)<106ε−2m_{|\cdot|}(\boldsymbol\theta)<10^6\varepsilon^{-2}, while Proposition 3 confirms that the order ε−2\varepsilon^{-2} is the appropriate scale.

02Proofs1 reported findingCorrect

The general construction theorem is proved in full, and its deductions of Theorems 1 and 2 are correct and complete.

Sections 6–11Correct and complete

Every induction condition is propagated

Pages 7–22 · proof of Theorem 3 and deductions · arXiv:2111.12259v1

The admissible annuli remain nonempty, the selected primitive vector satisfies the volume interval, and the accompanying planes contain exactly the stated integer points. The convergence of normalized directions produces a vector whose best approximations are precisely the constructed sequence, so the limiting spectral calculations are justified.

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:2111.12259v1
Authors listed
Renat K. Akhunzhanov, 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.