arXiv:0906.1541v2

On Kleinbock's Diophantine result

Nikolay G. Moshchevitin

math.NT11J13

Paper metadata

Original submission: June 8, 2009
Latest revision: January 28, 2011 · version 2
Claimed MSC: 11J13
arXiv categories: math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: Minor corrections, submitted to Publ. Math. Debrecen

Abstract

We give an elementary proof of a recent metrical Diophantine result by D. Kleinbock related to badly approximable vectors in affine subspaces.

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Audited against arXiv v2

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.

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Detailed mathematical audit

Generated August 20, 2026

01Statements3 reported findingsCorrect

The metric theorem is correct under its intended finite-scaling hypotheses. Corollary 2 needs the same logarithmic parameter restriction already used in Corollary 1.

Theorem 1Correct

Metric transfer from a badly approximable subspace

Page 2 · Theorem 1 · arXiv:0906.1541v2

The covering estimate gives a cumulative lattice-point count $\sum_{j\leq T}\zeta_j^{(R)}\ll\mu_T$ . Partial summation against $((\varphi(T))/T)^a$ and the assumed convergence of $\sum\mu_T\lambda_T$ make the exceptional limsup set null, proving that almost every point of $A$ is $\varphi$ -badly approximable.

Condition (5)Typo

The scaling supremum must be finite

Page 2 · Equation (5) and Theorem 1 · arXiv:0906.1541v2

Replace $\sup_{T\geq1}\varphi(\kappa T)/\varphi(T)\leq+\infty$ by $\sup_{T\geq1}\varphi(T)/\varphi(\kappa T)<+\infty$ , equivalently a positive lower bound for $\varphi(\kappa T)/\varphi(T)$ for each fixed $\kappa\geq1$ . The printed comparison with $+\infty$ is vacuous, while the Borel–Cantelli reduction needs exactly this fixed-scale lower comparison when $|x|\leq R|q|$ . The surrounding analogous condition (4) and the use on page 4 determine the correction.

Corollary 2Minor formal correction

The range $\Delta>1/a$ and the exponent $1/a$ should be displayed

Page 2 · Corollary 2 · arXiv:0906.1541v2

State $\Delta>1/a$ and define $\varphi_{a,\Delta}(T)=T^{-1/a}(\log T)^{-\Delta}$ . Equation (7) is written with exponent $1/d$ for Corollary 1, so merely referring to it leaves the $a$ -dimensional specialization and its necessary range unstated. With this local substitution, $\mu_T\lambda_T\asymp T^{-1}(\log T)^{-a\Delta}$ and the series converges exactly in the displayed range. No later statement depends on a different range.

02Proofs2 reported findingsCorrect

The lattice covering, Borel–Cantelli, and partial-summation steps are correct after the explicit scaling-notation correction.

Equation (9)Correct and complete

Covering count for integer points

Pages 2–3 · construction of $\Omega_T$ , $\Pi_T$ , and Equation (9) · arXiv:0906.1541v2

After ordering two lattice points by their zeroth coordinate, their difference lies in $\Omega_T$ . Thus every translate of $\frac12\Omega_T$ contains at most one lattice point. Comparing the $a-b$ transverse widths of $\Pi_T$ and $\Omega_T$ yields $O((T/\psi(T))^{a-b})=O(\mu_T)$ translates.

Borel–Cantelli reductionCorrect and complete

The exceptional sets have summable measure

Pages 3–4 · Equations (11)–(12) · arXiv:0906.1541v2

Each height- $T$ lattice point contributes an $a$ -dimensional ball of radius comparable to $\varphi(T)/T$ . Dyadic decomposition of Equation (9) gives $\sum_{j\leq T}\zeta_j^{(R)}\ll\mu_T$ , since monotonicity of $\psi$ supplies a geometric factor $2^{-\nu(a-b)}$ . Abel summation then converts the series to $\sum\mu_T\lambda_T$ , with the boundary term tending to zero.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:0906.1541v2
Authors listed: Nikolay G. Moshchevitin
Audit date: August 20, 2026

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On Kleinbock's Diophantine result

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Metric transfer from a badly approximable subspace

The scaling supremum must be finite

The range Δ>1/a\Delta>1/aΔ>1/a and the exponent 1/a1/a1/a should be displayed

Covering count for integer points

The exceptional sets have summable measure

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The range $\Delta>1/a$ and the exponent $1/a$ should be displayed