Published paper
Modified Schmidt games and a conjecture of Margulis
MSC 2020
11J8311J5437A1737A4637A44
Abstract
Proposition 3.1 packages the rank-one cusp-vector discreteness, compactness, and uniqueness properties quoted as Shi's Lemma 3.4.
Role in dependence graphs
Proof-critical source
Bounded trajectories of quasi-rays
This paper is included only for the following marked statement:
- Proposition 3.1 · source pp. 7–9Constructs a finite cusp-vector family satisfying discreteness, a precompactness criterion, and uniqueness of a sufficiently short vector; Shi quotes it as Lemma 3.4.
Proof-critical source
Submanifold-genericity of $\mathbb{R}^d$-actions and uniform multiplicative Diophantine approximation
This paper is included only for the following marked statement:
- Proposition 3.1 · source pp. 7–9Finite cusp-vector family with discreteness, precompactness, and uniqueness of a sufficiently short vector.
AI-generated audit
Audit summary
Not a correctness certificate. These reports do not replace expert scrutiny or formal verification.
Exact reviewed source
arXiv:1001.5017v2 · explicit fallback for version of record
Dmitry Kleinbock, Barak Weiss. Modified Schmidt games and a conjecture of Margulis. Journal of Modern Dynamics 7(3), 429–460 (2013). Reviewed form: arXiv:1001.5017v2.
The dependence graph cites the published article, but the fixed arXiv source is the proof-bearing form previously audited in the corpus. This report reuses that exact source audit and does not claim to audit the version of record.
Open audited source ↗01Statements2 reported findingsCorrect
The modified-game winning results for bounded and orbit-avoidance sets on homogeneous spaces, and the resulting thickness statement related to Margulis's conjecture, are correct.
Theorems 1.1 and 1.2Correct
The unstable-horospherical winning statements are correct
Pages 3–4 and 8–22 · Theorems 1.1–1.2 · arXiv:1001.5017v2
The admissible expanding region supplies a modified Schmidt game on the unstable subgroup. Quantitative non-divergence and the local product structure yield strategies for bounded forward orbits and for avoiding a prescribed countable collection of submanifolds.
Full paper, version 2 ↗Corollary 1.3 and Theorem 1.4Correct
The global thickness consequences follow from the slice results
Pages 4–5 and 22–33 · global consequences · arXiv:1001.5017v2
Winning gives full dimension on unstable leaves, while stable and neutral directions are restored through local product coordinates and a slicing theorem. The hypotheses on the centralizer and the orbit-closure obstruction are exactly those used in this passage.
02Proofs2 reported findingsCorrect
The modified-game, non-divergence, and local-product arguments are correct and complete.
Theorem 4.2Correct and complete
The bounded-orbit game strategy is complete
Pages 11–20 · Theorem 4.2 and proof · arXiv:1001.5017v2
At every game scale quantitative non-divergence bounds the dangerous cusp set, and the expanding subgroup action leaves a legal translate outside it. Iterating over a compact exhaustion yields a uniformly bounded trajectory.
Sections 6–8Correct and complete
Orbit avoidance and the Margulis application close
Pages 21–33 · Theorem 6.1 and final applications · arXiv:1001.5017v2
Transversality controls the intersections with forbidden submanifolds, countable intersections preserve winning, and local product coordinates transfer the leafwise dimension to the full homogeneous space without losing a dimension.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.