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Paper metadata

Original submission
December 21, 2008
Latest revision
September 23, 2009 · version 3
Claimed MSC
11J20
arXiv categories
math.NT
Comments
correction of misprints, more general result

Abstract

We give a sketch for an alternative proof of a recent result by J. Tseng.

AI-generated audit

Audit summary

Audited against arXiv v3

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 winning-set conclusions for badly approximable affine forms and their stated generalization are correct; the proof of the lacunary-hyperplane lemma needs the verified scheduling repair recorded below.

Theorem 1Correct

Winning property for affine badly approximable sets

Pages 2 and 4–6 · Theorem 1 and Lemma 1 · arXiv:0812.3998v3

After the lacunary normals are scheduled by their actual size bands, Schmidt's escaping lemma gives one game move for every finite band and hence a uniform positive lower bound for all defining affine forms. The transference construction preceding Lemma 1 supplies the required lacunary integer normals, so the set B(Θ)B(\Theta) is α\alpha-winning for every 0<α<1/20<\alpha<1/2.

Theorem 2Correct

The approximation-function generalization

Page 3 and pages 4–6 · Theorem 2 and proof scheme · arXiv:0812.3998v3

Under ψΘ(t)≤ψ(t)\psi_\Theta(t)\leq\psi(t), the inverse scale ρ\rho converts the selected dual approximations into the same lacunary family used for Theorem 1. Applying the repaired size-band game argument at those scales gives a uniform positive lower bound for max⁡j∥Lj(x)−ηj∥ ρ(max⁡i∣xi∣)\max_j\lVert L_j(x)-\eta_j\rVert\,\rho(\max_i|x_i|), which is exactly the stated winning set.

Theorem 2 notationTypo

The size of the integer vector uses the wrong bars

Page 3 · statement of Theorem 2 · arXiv:0812.3998v3

The argument of the decreasing function is printed using the distance-to-the-nearest-integer notation on integer coordinates. Replace it by the sup-norm max⁡i∣xi∣\max_i |x_i|. This is uniquely determined by the preceding definition of the approximation set and does not change the theorem.

02Proofs1 reported findingContains incorrect or incomplete proofs

The proof imposes an upper lacunarity bound that is not a consequence of the lemma's hypotheses. A verified size-band scheduling argument repairs the proof without changing either theorem.

Lemma 1Incomplete as written · verified repair

The claimed upper lacunarity reduction is unjustified

Page 5 · proof of Lemma 1, reduction to tr+1/tr≤M2t_{r+1}/t_r\leq M^2 · arXiv:0812.3998v3

Lemma 1 assumes only tr+1/tr≥M>1t_{r+1}/t_r\geq M>1, but the proof says without justification that one may also impose tr+1/tr≤M2t_{r+1}/t_r\leq M^2 and then uses that upper bound to synchronize the normals with game scales. Verified repair: at each game scale, group the normals into the half-open bands for which the product of the current ball radius and trt_r lies in a fixed interval. The lower lacunarity bound gives a uniform bound, depending only on MM, for the number of normals in each band; Schmidt's escaping lemma removes that finite band in a fixed number of moves. Every normal enters exactly one band, so the resulting point has a uniform positive distance from all corresponding affine hyperplanes. This supplies the lemma under its printed one-sided lacunarity hypothesis and leaves all downstream constants qualitative only.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:0812.3998v3
Authors listed
Nikolay G. Moshchevitin
Audit date
August 20, 2026
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