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arXiv:0904.1614v2

An `almost all versus no' dichotomy in homogeneous dynamics and Diophantine approximation

Dmitry Kleinbock

math.DSmath.NT11J1337A17
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Paper metadata

Original submission
April 9, 2009
Latest revision
August 6, 2009 · version 2
Claimed MSC
11J13, 37A17
arXiv categories
math.DS, math.NT
Comments
14 pages; section 3.3 added
Journal reference
Geom. Dedicata 149 (2010), 205-218

Abstract

Let Y0Y_0 be a not very well approximable m×nm\times n matrix, and let MM be a connected analytic submanifold in the space of m×nm\times n matrices containing Y0Y_0. Then almost all Y∈MY\in M are not very well approximable. This and other similar statements are cast in terms of properties of certain orbits on homogeneous spaces and deduced from quantitative nondivergence estimates for `quasi-polynomial' flows on on the space of lattices.

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Audit summary

Audited against arXiv v2

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.

Current report

Detailed mathematical audit

Generated August 19, 2026
01Statements2 reported findingsCorrect

The almost-all-versus-none dichotomy for analytic manifolds and the corresponding orbit dichotomy in homogeneous space are correct.

Theorem 1.4Correct

Analytic-manifold Diophantine dichotomy

Pages 3–5 · Theorem 1.4 · arXiv:0904.1614v2

Analyticity makes each exterior-covolume function either identically constrained on the connected parameter domain or uniformly good away from its zero set. Quantitative nondivergence then forces every improved-approximation limsup set to have either full measure or zero measure, and connectedness makes the alternative global.

Full paper, version 2 ↗
Theorem 1.7Correct

Homogeneous orbit dichotomy

Pages 5–6 and Sections 2–4 · Theorem 1.7 · arXiv:0904.1614v2

For an analytic family of unipotent translates, every rational exterior vector has an analytic norm function. The same identically-small versus quantitative-lower-bound alternative yields that almost every orbit has the stated recurrence property or none does.

02Proofs2 reported findingsCorrect

The analytic good-function lemma, quantitative nondivergence estimate, and dynamical correspondence are correct and complete.

Theorem 2.3Correct and complete

Quantitative nondivergence covers every primitive subgroup

Section 2 · Theorem 2.3 · arXiv:0904.1614v2

Analytic coordinate functions are uniformly good on compact subballs unless identically zero. Applying the nondivergence theorem to the finite-rank family of primitive subgroups gives a summable exceptional estimate whenever the algebraic obstruction is absent.

Sections 3–4Correct and complete

Connected analyticity makes the dichotomy exhaustive

Sections 3–4 · proofs of Theorems 1.4 and 1.7 · arXiv:0904.1614v2

An exterior coefficient vanishing on a positive-measure analytic subset vanishes identically on the connected domain. Otherwise its uniform lower bound on a smaller ball feeds the nondivergence estimate. The Dani correspondence transfers these mutually exclusive alternatives to the exact Diophantine and orbit statements.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:0904.1614v2
Authors listed
Dmitry Kleinbock
Audit date
August 19, 2026
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