Published paper
Abstract
Lemma 3.2 proves the multivariable polynomial-good estimate used for the exterior-power covolume functions in Shi's arithmetic nondivergence argument.
Role in dependence graphs
Proof-critical source
Bounded trajectories of quasi-rays
This paper is included only for the following marked statement:
- Lemma 3.2 · source p. 8Shows that every degree- polynomial on is -good, verifying the good-function hypothesis in Shi's arithmetic nondivergence branch.
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:
- Lemma 3.2 · source p. 8Polynomial-good estimate used for Shi's exterior-power covolume functions.
AI-generated audit
Audit summary
Not a correctness certificate. These reports do not replace expert scrutiny or formal verification.
Exact reviewed source
arXiv:math/0210298v1 · explicit fallback for version of record
Victor Bernik, Dmitry Kleinbock, G. A. Margulis. Khintchine-type theorems on manifolds: the convergence case for standard and multiplicative versions. International Mathematics Research Notices 2001(9), 453–486 (2001). Reviewed form: arXiv:math/0210298v1.
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 convergence Khintchine-type theorem for nondegenerate manifolds, including its multiplicative form, is correct under the stated smoothness and summability hypotheses.
Theorem 1.1Correct
The convergence theorem has the correct scope
Pages 2–4 · Theorem 1.1 · arXiv:math/0210298v1
The nondegeneracy condition supplies uniform control outside a null exceptional set, and the displayed convergence series is the one required after decomposing integer coefficients into dyadic blocks. The standard and multiplicative conclusions retain the correct almost-everywhere quantifiers.
Full paper, version 1 ↗Theorems 1.3 and 1.4Correct
The large- and small-gradient estimates support the main theorem
Pages 5–9 · Theorems 1.3–1.4 · arXiv:math/0210298v1
The large-gradient part is controlled by a change-of-variables estimate, while the small-gradient part is reduced to quantitative non-divergence for the associated lattices. Their parameter ranges cover the dyadic decomposition used in Theorem 1.1.
02Proofs2 reported findingsCorrect
The gradient decomposition, quantitative non-divergence argument, and final Borel-Cantelli summation are correct and complete.
Large-gradient argumentCorrect and complete
The local measure estimate is complete
Pages 9–13 · proof of Theorem 1.3 · arXiv:math/0210298v1
A coordinate with uniformly large derivative is selected on each local chart, and Fubini together with one-dimensional monotonicity gives the required measure bound with constants uniform over each dyadic block.
Small-gradient and summation argumentCorrect and complete
Quantitative non-divergence gives a summable exceptional set
Pages 13–27 · proof of Theorem 1.4 and completion of Theorem 1.1 · arXiv:math/0210298v1
The exterior-power lower bounds required by quantitative non-divergence follow from nondegeneracy, and the resulting estimates remain summable after the multiplicative dyadic partition. Borel-Cantelli then yields exactly the claimed null set.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.