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arXiv:math/0508517v3

An extension of quantitative nondivergence and applications to Diophantine exponents

Dmitry Kleinbock

math.DSmath.NT22F3011J83
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Paper metadata

Original submission
August 25, 2005
Latest revision
May 19, 2008 · version 3
Claimed MSC
22F30, 11J83
arXiv categories
math.DS, math.NT
Comments
26 pages, a revised version. Trans. AMS, to appear

Abstract

We present a sharpening of nondivergence estimates for unipotent (or more generally polynomial-like) flows on homogeneous spaces. Applied to metric Diophantine approximation, it yields precise formulas for Diophantine exponents of affine subspaces of Rn\Bbb R^n and their nondegenerate submanifolds.

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Audited against arXiv v3

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

Generated August 19, 2026
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 ωj(A)\omega_j(A). 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.

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Paper
arXiv:math/0508517v3
Authors listed
Dmitry Kleinbock
Audit date
August 19, 2026
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