arXiv:0805.2934v2
Modified Schmidt games and Diophantine approximation with weights
math.NTmath.DS11J8328A80
Abstract
We show that the sets of weighted badly approximable vectors in are winning sets of certain games, which are modifications of -games introduced by W. Schmidt in 1966. The latter winning property is stable with respect to countable intersections, and is shown to imply full Hausdorff dimension.
AI-generated audit
Audit summary
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
01Statements2 reported findingsCorrect
The modified-Schmidt-game winning theorem for weighted badly approximable vectors and its intersection consequences are correct; two displayed notation errors are harmless typos.
Theorem 1.1Correct
Weighted badly approximable vectors are winning for the stated game
Pages 3 and 17–24 · Theorem 1.1 and its proof · arXiv:0805.2934v2
The diagonal contraction assigns the required scale to each weighted coordinate, and the resonant rational points at a fixed scale lie in an affine hyperplane that can be avoided by one legal move. The countable-intersection and thickness conclusions then follow from the established game axioms.
Full paper, version 2 ↗Equations (1.2) and (1.6)Typos · no status impact
The weight index and one closing delimiter are typographical errors
Pages 2–3 · equations (1.2) and (1.6) · arXiv:0805.2934v2
The weight vector has entries indexed from one through n, but its normalization sum ends at an undefined m; the unique correction is n. The final diagonal entry in the displayed flow also has one unmatched closing parenthesis; deleting that parenthesis gives the expression used throughout the proof.
02Proofs2 reported findingsCorrect
The abstract game results, geometric contraction estimates, and resonant-set avoidance proof are correct and complete after the displayed typo corrections.
Sections 2–3Correct and complete
The modified-game framework has the required stability properties
Pages 4–15 · abstract modified Schmidt games · arXiv:0805.2934v2
The nesting and diameter axioms guarantee a unique outcome, the strategy-splicing argument proves closure under countable intersections, and the measure hypothesis yields full Hausdorff dimension of winning sets.
Section 4Correct and complete after the stated repairs
The weighted resonant-set strategy closes
Pages 16–26 · proof of Theorem 1.1 · arXiv:0805.2934v2
The simplex-type lemma confines relevant rational points to a hyperplane at each scale, and the contraction parameters leave enough room to avoid that hyperplane. Iteration gives the uniform weighted lower bound defining the target set.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.