arXiv:1906.00747v2
Quantitative non-divergence and Diophantine approximation on manifolds
math.NTmath.DS11JXX37A17
Abstract
The goal of this survey is to discuss the Quantitative non-Divergence estimate on the space of lattices and present a selection of its applications. The topics covered include extremal manifolds, Khintchine-Groshev type theorems, rational points lying close to manifolds and badly approximable points on manifolds. The main emphasis is on the role of the Quantitative non-Divergence estimate in the aforementioned topics within the theory of Diophantine approximation, and therefore this paper should not be regarded as a comprehensive overview of the area.
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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.
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Detailed mathematical audit
01Statements3 reported findingsCorrect
The survey's formulations of quantitative non-divergence and its selected Diophantine applications are correct and accurately scoped to their cited sources; one citation label contains a spelling typo.
Theorems 1–5Correct
The non-divergence and metric-approximation statements are correct
Pages 3–15 · Theorems 1–5 · arXiv:1906.00747v2
The goodness and lower-bound hypotheses in quantitative non-divergence are retained, and the extremality and convergence Khintchine-Groshev consequences are stated with the required nondegeneracy, smoothness, and monotonicity assumptions.
Full paper, version 2 ↗Theorems 6–10Correct
The dimension, rational-point, and badly-approximable applications are correctly presented
Pages 16–31 · Theorems 6–10 · arXiv:1906.00747v2
Each theorem is identified as a cited result, and the survey preserves the hypotheses needed for the stated dimension bounds, rational-point estimates, and winning conclusions. Open problems are clearly separated from proved statements.
Citation in Theorem 1Typo · no status impact
The word Theorem is misspelled in the source locator
Page 3 · heading of Theorem 1 · arXiv:1906.00747v2
The parenthetical citation says Theprem 5.2. Replacing that word by Theorem gives the intended source locator and has no mathematical effect.
02Proofs2 reported findingsCorrect
At the paper's declared survey level, the derivations are correct and every imported theorem is explicitly attributed to a verifiable source.
Quantitative non-divergence applicationsCorrect and complete for a survey
The lattice reductions use the cited theorem with all hypotheses visible
Pages 3–15 · Sections 1–3 · arXiv:1906.00747v2
The paper explains how nondegeneracy gives goodness, how exterior-product lower bounds prevent collapse, and how Borel-Cantelli turns the measure estimate into extremality and convergence conclusions. The omitted full proofs are explicitly assigned to the cited original papers.
Later applicationsCorrect and complete at the stated survey level
The explanatory deductions do not overstate the cited results
Pages 16–32 · Sections 4–6 · arXiv:1906.00747v2
The dimension and rational-point applications are derived only where the quoted estimates apply, and the badly approximable section separates proved winning results from conjectural extensions. No uncited proof claim is used to enlarge a theorem's scope.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.