Published paper
An extension of quantitative nondivergence and applications to Diophantine exponents
MSC 2020
37A1711J83
Abstract
Theorem 2.2 gives the Besicovitch–Federer quantitative nondivergence estimate that proves Shi's arithmetic bad-set bound (3.5).
Role in dependence graphs
Proof-critical source
Bounded trajectories of quasi-rays
This paper is included only for the following marked statement:
- Theorem 2.2 · source pp. 9–10Quantitative nondivergence for under good covolume functions and uniform lower bounds; it yields Shi's estimate (3.5).
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:
- Theorem 2.2 · source pp. 9–10Quantitative nondivergence under good covolume functions and uniform lower bounds.
AI-generated audit
Audit summary
Not a correctness certificate. These reports do not replace expert scrutiny or formal verification.
Exact reviewed source
arXiv:math/0508517v3 · explicit fallback for version of record
Dmitry Kleinbock. An extension of quantitative nondivergence and applications to Diophantine exponents. Transactions of the American Mathematical Society 360(12), 6497–6523 (2008). Reviewed form: arXiv:math/0508517v3.
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 strengthened quantitative nondivergence result, inheritance of Diophantine exponents by nondegenerate submanifolds of affine subspaces, and the explicit affine-subspace exponent formulas are correct.
Theorem 0.2Correct
Quantitative nondivergence with a partially ordered index set
Introduction and Section 2 · Theorem 0.2 · arXiv:math/0508517v3
The poset formulation tracks the primitive subgroups that can become small at a point. Federer doubling, Besicovitch covering, goodness, and the uniform lower bound give the announced measure estimate through induction on chains; the argument does not require a simultaneous lower bound at every point.
Full paper, version 3 ↗Theorems 0.3 and 0.4Correct
Inheritance and affine-subspace exponent formulas
Introduction and Sections 4–5 · Theorems 0.3–0.4 · arXiv:math/0508517v3
The dynamical criterion reduces exponents to growth rates of primitive exterior vectors. Nondegeneracy makes the relevant coefficient functions good and prevents extra rational relations, so the ambient affine subspace and its nondegenerate submanifold have the same exponent. Computing the finitely many higher-order exponents of the parametrizing matrix yields the displayed formula.
02Proofs2 reported findingsCorrect
The poset nondivergence proof, exterior-algebra reduction, and higher-order exponent calculation are correct and complete.
Theorems 2.1–2.2Correct and complete
The covering induction handles every primitive rank
Section 2 · Theorems 2.1–2.2 · arXiv:math/0508517v3
At each point only a chain of marked indices can occur, so its length is uniformly bounded. The Besicovitch subcover and Federer estimate reduce the exceptional measure at one rank, and induction gives the stated constant and exponent over the entire ball.
Sections 3–5Correct and complete
Exterior powers give the claimed inheritance formula
Sections 3–5 · arXiv:math/0508517v3
Dani's correspondence is expressed for every primitive rank, and the coefficient expansion isolates precisely the higher-order exponents . The proof establishes both directions of the maximum formula and then transfers it to every nondegenerate submanifold by the good/nonplanar criterion.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.