MathAudit

Paper metadata

Original submission
December 31, 2025
Claimed MSC
11H16, 11H46, 11J13
arXiv categories
math.NT
Comments
15 pages

Abstract

We describe the spectrum of ordinary Diophantine exponents for dd-dimensional lattices. The result reduces the problem to two-dimensional case and uses argument of metric theory.

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 spectrum of lattice Diophantine exponents is [0,+∞][0,+\infty] in every dimension at least three, and the quantitative exact-order theorem supports this conclusion.

Theorems 1–2Correct

The exact-order construction realizes every nonnegative exponent

Pages 2–3 and 13–15 · Theorems 1–2 and Section 8 · arXiv:2512.24913v1

The prescribed function supplies infinitely many lattice points at the upper target scale while the two Borel–Cantelli estimates exclude all but finitely many points below the lower scale. Taking w(t)=t−γdw(t)=t^{-\gamma d} therefore realizes the exponent γ\gamma.

Full paper, version 1 ↗
Condition (C) and the final case splitTypos · no status impact

Two unbound or shifted indices are typos

Pages 13–14 · condition (C) and cases 3–5 · arXiv:2512.24913v1

Condition (C) must compare the ν\nuth convergent with w~(qν)\widetilde w(q_\nu), not with an unbound qνkq_{\nu_k}. In case 3, qν+kq_{\nu+k} must be qνkq_{\nu_k}. The surrounding equalities uniquely fix both indices.

02Proofs2 reported findingsContains incorrect or incomplete proofs

The geometric and metric parts of the construction are correct, but the cited exact-approximation theorem is used with an unproved localization requirement. The gap has a standard verified repair by fixing an initial continued-fraction cylinder.

Section 8, choice of alphaIncomplete as written; verified repair

The quoted Baker–Ward theorem does not by itself place alpha near the required point

Pages 3 and 13 · Theorem B and conditions (A)–(C) · arXiv:2512.24913v1

Condition (A) requires ∣α−1/4∣≤δ|\alpha-1/4|\leq\delta, whereas the quoted Theorem B only asserts an uncountable set somewhere in [0,1][0,1]. The proof gives no density or localization statement for that set. A repair is to prescribe a finite initial continued-fraction word whose cylinder lies in the desired interval and apply the Baker–Ward tail construction after that word; finite prefixes alter neither the eventual lower bound nor the infinitely many exact-order convergents. With this localization inserted, the rest of Section 8 applies unchanged.

Sections 4–7Correct and complete

The two Borel–Cantelli arguments and embedding estimates close

Pages 5–13 · First and Second Main Lemmas · arXiv:2512.24913v1

The planar exceptional sets have summable measure, the multidimensional boxes have the required product-volume bounds, and the coefficient count makes the second exceptional series converge under (4). These conclusions are uniform over the compact parameter box used in the final lattice construction.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:2512.24913v1
Authors listed
Nikolay Moshchevitin
Audit date
August 20, 2026
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