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Paper metadata

Original submission
March 26, 2026
Claimed MSC
11J04, 11J70, 11J13
arXiv categories
math.NT
Comments
14 pages

Abstract

We study properties of Diophantine exponents of lattices and so-called related "weak" uniform approximations introduced in recent papers by Oleg German, in the simplest two-dimensional case. In contrast to the multidimensional case, in the two-dimensional case we can use a powerful tool of continued fractions. We develop an analog of Jarník's theory dealing with inequalities between the ordinary and uniform Diophantine exponents, which turned out to be related to mutual behaviour of irrationality measure functions for two real numbers.

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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 findingsContains wrong statements

The inequalities for one number and for the two-number exponents are correct, but the lattice corollary printed as Theorem 4 is false. The paper's own sharpness family gives a counterexample to it.

Theorems 1–3Correct

The one- and two-number exponent bounds are supported

Pages 4–12 · Theorems 1–3 and Sections 3–4 · arXiv:2603.25071v1

The piecewise-constant comparison lemma supplies the necessary alternating growth inequalities, and the continued-fraction estimates translate those inequalities into the stated ordinary exponent bounds. The family in Section 4.3 attains the Theorem 3 relation asymptotically.

Full paper, version 1 ↗
Theorem 4Incorrect

The stated lattice inequality is contradicted by Section 4.3

Pages 6 and 12 · Theorem 4, formula (15), and Section 4.3 · arXiv:2603.25071v1

Let the parameter γ\gamma in the family of Section 4.3 tend to zero. That construction has ϖυ=1+γ\varpi_\upsilon=1+\gamma and ω(θ)=ω(η)=g\omega(\theta)=\omega(\eta)=g, where (g−1)2=γ2g(g-1)^2=\gamma^2g; hence formula (15) gives both lattice exponents tending to 11. The printed right-hand side of Theorem 4 instead tends to 1+21+\sqrt2. The correct direct consequence of Theorem 3 and (15) is ωΛ≥g2ωˉΛ−1+12,\omega_\Lambda\geq\frac{g_{2\bar\omega_\Lambda-1}+1}{2}, with gyg_y the root defined in (17).

02Proofs2 reported findingsContains incorrect or incomplete proofs

The proofs of Theorems 1–3 are complete, but the derivation of Theorem 4 uses an invalid algebraic transformation. A corrected lattice corollary is explicitly determined by formula (15).

Derivation of Theorem 4Incorrect as written; corrected statement verified

The passage from the two-number exponent to the lattice exponent is algebraically invalid

Pages 4–6 · formulas (15), (17), and Theorem 4 · arXiv:2603.25071v1

Writing ωˉΛ=(ϖυ+1)/2\bar\omega_\Lambda=(\varpi_\upsilon+1)/2 and ωΛ=(max⁡(ω(θ),ω(η))+1)/2\omega_\Lambda=(\max(\omega(\theta),\omega(\eta))+1)/2 in Theorem 3 yields the corrected bound displayed in the statement finding. It does not yield the newly defined root in (20) multiplied by ωˉΛ\bar\omega_\Lambda. The internal sharpness family formally disproves that latter expression.

Proof of Theorem 1Typo · no additional status impact

The endpoint scale is introduced in the wrong order

Page 5 · proof of Theorem 1 · arXiv:2603.25071v1

The proof first defines ν∗\nu_* from qν∗≤t<qν∗+1q_{\nu_*}\leq t<q_{\nu_*+1} and only afterwards sets t=qν∗−1t=q_{\nu_*}-1, which is inconsistent with that definition. Choose a sufficiently large index ν∗\nu_* first and then set t=qν∗−1t=q_{\nu_*}-1. The following minimum and recurrence estimates then apply as printed, so this local ordering error does not affect Theorem 1.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:2603.25071v1
Authors listed
Nikolay Moshchevitin
Audit date
August 20, 2026
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