Published paper
Flows on homogeneous spaces and Diophantine approximation on manifolds
MSC 2020
11J1311J8322E9957S25
Abstract
Proposition 3.2 and Lemma 3.3 prove the one-variable and multivariable good-function estimates inherited by Bernik–Kleinbock–Margulis.
Role in dependence graphs
Proof-critical source
Bounded trajectories of quasi-rays
This paper is included only for the following marked statement:
- Proposition 3.2 and Lemma 3.3 · source pp. 6–9Proves the one-variable polynomial-good estimate and the multivariable induction used in Bernik–Kleinbock–Margulis Lemma 3.2.
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 3.2 and Lemma 3.3 · source pp. 6–9One-variable polynomial-good bound and multivariable induction behind BKM Lemma 3.2.
AI-generated audit
Audit summary
Not a correctness certificate. These reports do not replace expert scrutiny or formal verification.
Exact reviewed source
arXiv:math/9810036v1 · explicit fallback for version of record
Dmitry Kleinbock, G. A. Margulis. Flows on homogeneous spaces and Diophantine approximation on manifolds. Annals of Mathematics 148(1), 339–360 (1998). Reviewed form: arXiv:math/9810036v1.
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 strong-extremality theorem for nondegenerate manifolds and its metric Diophantine consequences are correct.
Theorem ACorrect
Nondegenerate manifolds are strongly extremal
Pages 3--17 · Theorem A, its reduction, and proof of Proposition 2.3 · arXiv:math/9810036v1
Very-well multiplicative approximation produces a multiparameter diagonal orbit with shortest-vector function decaying exponentially in total time. At a nondegenerate parameter point, finite-order derivatives make every relevant linear combination uniformly good, while the exterior-power lower bounds prevent a primitive subgroup from remaining uniformly small. Quantitative nondivergence then gives exponentially summable exceptional-set measures, and Borel--Cantelli proves strong extremality.
Theorem BCorrect
The convergence Khintchine-type conclusion has the stated range
Pages 17--18 · Theorem B and its deduction from quantitative nondivergence · arXiv:math/9810036v1
The decreasing approximation function is grouped into dyadic height blocks. The theorem's series condition makes the corresponding quantitative-nondivergence bounds summable. Nondegeneracy supplies the same uniform goodness and subgroup lower bounds as in Theorem A, so almost every parameter belongs to only finitely many approximation blocks.
02Proofs2 reported findingsCorrect
The lattice correspondence, good-function estimates, quantitative nondivergence theorem, and Borel--Cantelli summations are correct and complete.
Section 2Correct and complete
The multiplicative approximation-to-cusp reduction has the correct exponent
Pages 4--5 · Lemma 2.1, Corollary 2.2, and Proposition 2.3 · arXiv:math/9810036v1
Choosing and gives and hence both the linear-form and coordinate components are at most . Eliminating gives the positive exponential rate . Rounding the times changes the shortest-vector bound by only a fixed factor.
Sections 3--5Correct and complete
Quantitative nondivergence covers every primitive subgroup
Pages 6--17 · good functions and Theorem 5.2 · arXiv:math/9810036v1
Finite-order nondegeneracy gives a uniform sublevel exponent for every real linear combination of . The marked-point induction controls all ranks of primitive subgroups, not only short vectors, and the lower-bound hypothesis is verified using the nonvanishing derivative span. Substituting an exponentially small threshold yields the summable estimate used in the main proof.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.