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Paper metadata

Original submission
February 22, 2011
Claimed MSC
11J13
arXiv categories
math.NT
Comments
6 pages

Abstract

Consider irrational affine subspace A⊂Rd A\subset \mathbb{R}^d of dimension aa. We prove that the set {ξ=(ξ1,...,ξd)∈A:    q1/a⋅max⁡1≤i≤d∣∣qξi∣∣→∞,    q→∞} \{ξ=(ξ_1,...,ξ_d) \in {A}:\,\,\, \ q^{1/a}\cdot \max_{1\le i \le d} ||qξ_i|| \to \infty,\,\,\,\, q\to \infty\} is an αα-winning set for every α∈(0,1/2]α\in (0,1/2]

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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 findingsCorrect

The Jarnik-type winning theorem for irrational affine subspaces is correct after two mechanically determined subscript corrections in its auxiliary lemmas.

Theorem 4Correct

Diverging normalized approximation is winning

Pages 2 and 5-6 · Theorem 4 and Section 4 · arXiv:1102.4431v1

At every game scale the rational points of bounded denominator lie in one proper affine subspace. Schmidt's escaping lemma lets White avoid that subspace and the next rational obstruction. The determinant thresholds tend to infinity because dim⁡Γ(A)<a\dim\Gamma(A)<a, and substitution into (12) gives q1/amax⁡i∥qξi∥→∞q^{1/a}\max_i\|q\xi_i\|\to\infty.

Full paper, version 1 ↗
Lemma 2 thresholdTypo · no status impact

The determinant index is a typographical mismatch

Pages 3-4 · Lemma 2 and its proof · arXiv:1102.4431v1

After defining kk by νk≤n<νk+1\nu_k\leq n<\nu_{k+1}, each occurrence of dνnd_{\nu_n} in Lemma 2 and its proof must be dνkd_{\nu_k}. The exclusion of V1,…,VnV_1,\ldots,V_n gives D≥dνkD\geq d_{\nu_k}, and the theorem's later application explicitly uses dνkrd_{\nu_{k_r}}. The correction is unique and leaves the determinant argument unchanged.

02Proofs3 reported findingsCorrect

The rational-hyperplane lemma and game induction are complete; one false distance label is a harmless typo with a unique repair.

Lemma 1Typo · no status impact

The separating-hyperplane distance names the wrong subspace

Page 3 · proof of Lemma 1 · arXiv:1102.4431v1

The printed assertion dist⁡(L′′,U)>0\operatorname{dist}(L'',U)>0 is impossible because L′⊂L′′∩UL'\subset L''\cap U. It must read dist⁡(L′′,V)>0\operatorname{dist}(L'',V)>0: the preceding sentence constructs L′′L'' disjoint from VV, and this positive separation is exactly what places the chosen half of UU away from VV.

Lemma 2Correct and complete after the stated typo repair

The simplex-type determinant contradiction is valid

Pages 3-4 · Lemma 2 · arXiv:1102.4431v1

If a+1a+1 rational points were independent, their primitive lifts would span a completely rational (a+1)(a+1)-space with covolume at least dνkd_{\nu_k}. The cylinder-section volume is strictly below dνk/(a+1)!d_{\nu_k}/(a+1)!, contradicting the simplex volume of independent lattice points.

Section 4Correct and complete

The denominator-scale induction reaches the target limit

Pages 4-6 · equations (8)-(13) · arXiv:1102.4431v1

The two escaping moves give (10)-(11), hence the uniform lower bound (12) for all q<Rrq<R_r. The avoided rational subspaces force kr→∞k_r\to\infty; comparing qq with Rr−1R_{r-1} transfers this growth into the factor dνkr−21/ad_{\nu_{k_{r-2}}}^{1/a}, which proves the required divergence.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:1102.4431v1
Authors listed
Nikolay Moshchevitin
Audit date
August 20, 2026
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