Published paper

On effective equidistribution of expanding translates of certain orbits in the space of lattices

Dmitry Y. Kleinbock, G. A. Margulis

MSC 2020

37A1737A25

Published version ↗Published PDF ↗Open DOI ↗arXiv record →

Publication metadata

Journal citation: Number Theory, Analysis and Geometry, 385–396 (2012)
Publication date: 2012
DOI: 10.1007/978-1-4614-1260-1_18 ↗
arXiv counterpart: arXiv:math/0702433v4
MSC 2020: 37A17, 37A25
As specified in the publication (MSC 2010): 37A17, 37A25 ↗Springer primary chapter page, immediately after the abstractBoth codes carry forward unchanged. The publisher does not assign primary/secondary roles; none is inferred.
Official MSC conversion: MSC 2010 → MSC 2020 ↗

Abstract

Proposition 3.5 gives the quantitative injectivity-radius bound for shortest-vector compacta that Shi uses in the arithmetic branch of quantitative nonescape.

Role in dependence graphs

Proof-critical source

Bounded trajectories of quasi-rays

This paper is included only for the following marked statement:

Proposition 3.5 · source p. 8Bounds the injectivity radius of the shortest-vector compactum by $r(K_\varepsilon)\geq c(k)\varepsilon^k$ , converting Shi's lattice nondivergence estimate into the injectivity-radius form of Theorem 1.3.

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 3.5 · source p. 8Quantitative injectivity-radius lower bound for shortest-vector compacta.

AI-generated audit

Audit summary

Audited against an explicitly disclosed arXiv fallback

Not a correctness certificate. These reports do not replace expert scrutiny or formal verification.

Exact reviewed source

arXiv:math/0702433v4 · explicit fallback for version of record

Dmitry Y. Kleinbock, G. A. Margulis. On effective equidistribution of expanding translates of certain orbits in the space of lattices. Number Theory, Analysis and Geometry, 385–396 (2012). Reviewed form: arXiv:math/0702433v4.

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 ↗

Generated August 23, 2026

01Statements2 reported findingsCorrect

The effective equidistribution theorem for expanding translates of the matrix horosphere is correct. Several local notation and sign defects have uniquely determined repairs and do not change the theorem.

Theorem 1.3Correct

Equidistribution is exponentially effective uniformly away from the walls

Pages 2--3 and Section 4 · main theorem and proof · arXiv:math/0702433v4

Writing $g_{\mathbf t}=g_tg_{\mathbf u}$ with both $t$ and every coordinate of $\mathbf u$ comparable to $\lfloor\mathbf t\rfloor$ separates the argument into nondivergence under $g_{\mathbf u}$ and exponential mixing under $g_t$ . Quantitative nondivergence removes an exponentially small exceptional part, while smoothing the remaining horospherical pieces and applying exponential mixing gives another exponential error. Optimizing the smoothing and compactness parameters yields the asserted rate.

Uniformity in the base latticeCorrect

The compact-set dependence is handled correctly

Sections 3--4 · Corollary 3.4, Proposition 3.5, and proof of Theorem 1.3 · arXiv:math/0702433v4

The exterior-power expansion lemma gives a lower bound uniform over base lattices in a fixed compact set. Quantitative nondivergence therefore puts all but an exponentially small set in $K_\varepsilon$ . The injectivity radius bound $r(K_\varepsilon)\gg\varepsilon^k$ makes the local thickening legal uniformly, and the resulting Sobolev losses are absorbed by the chosen exponential scale.

02Proofs6 reported findingsCorrect

The nondivergence, injectivity-radius, smoothing, and mixing estimates are correct and complete after uniquely determined local notation and exponent-sign repairs.

Equation (1.2)Typo

The final contracting diagonal entry has the wrong sign

Page 1 · definition of the standard diagonal flow · arXiv:math/0702433v4

The matrix is declared to lie in $\operatorname{SL}_{m+n}(\mathbb R)$ and to have $n$ contracting entries, but its last printed entry is $e^{t/n}$ rather than $e^{-t/n}$ . Replacing the final sign by a minus makes the determinant one and matches every later use of $g_t$ .

Definition of $I_{f,\psi}$Typo

The defining integral uses an undefined time element instead of its group argument

Page 1 · display defining $I_{f,\psi}(g,z)$ · arXiv:math/0702433v4

The left side is $I_{f,\psi}(g,z)$ for an arbitrary $g\in G$ , but the integrand is printed as $\psi(g_t h z)$ , leaving $t$ undefined and $g$ unused. The following sentence calls this the $g$ -translate, and every subsequent specialization writes $I_{f,\psi}(g_t,z)$ ; therefore the definition must read $\psi(g h z)$ .

Theorem 2.1Typo

The universal quantifier names an unused time variable

Page 3 · exponential mixing theorem · arXiv:math/0702433v4

The theorem says `for any $t\geq0$ ' but its estimate contains an arbitrary group element $g$ and the factor $e^{-\gamma\operatorname{dist}(g,e)}$ . The cited result and the subsequent substitution $g=g_t$ require `for any $g\in G$ '. This replacement is uniquely determined and changes no application.

Proof of Theorem 2.3Typo

The base lattice is denoted by an undefined variable

Page 5 · thickening calculation in the proof of Theorem 2.3 · arXiv:math/0702433v4

The theorem fixes $z\in X$ and defines $\varphi(h^-h^0hz)$ , but the next three integrals contain $g_t h x$ and $g_t h^-h^0 h x$ although no $x$ was introduced. Replacing each $x$ by $z$ is forced by the definition of $\varphi$ and restores the displayed comparison with $I_{f,\psi}(g_t,z)$ .

Equations (2.7) and the following mixing estimateTypo

The smoothing-radius exponents have the wrong signs

Pages 5--6 · Sobolev estimates in the proof of Theorem 2.3 · arXiv:math/0702433v4

Lemma 2.2 gives a normalized bump supported at scale $r$ the bound $\|\theta\|_\ell\ll r^{-(\ell+N/2)}$ . Consequently equation (2.7) must read $\|\widetilde\theta\|_\ell\ll r^{-(2\ell+N/2)}$ , and the mixing estimate immediately below must carry the same negative exponent. The theorem's displayed bound (2.4), the intervening line for $\|\varphi\|_\ell$ , and the optimization in Remark 2.4 all already use the negative exponent, so both sign repairs are uniquely determined.

Section 4Correct and complete

The two error terms can be optimized simultaneously

Pages 8--10 · proof of Theorem 1.3 · arXiv:math/0702433v4

The bad-set measure is a positive power of the compactness threshold, while Sobolev and injectivity losses are fixed negative powers of that threshold and the smoothing radius. Choosing both as sufficiently small exponentials in $\lfloor\mathbf t\rfloor$ leaves a positive exponent in every term. The finite range of smaller $\lfloor\mathbf t\rfloor$ is absorbed into the constant.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

Detailed audit reportFull reasoning, exact locations, and sources checked.

Open report PDF ↗