arXiv:1710.04898v2

Dimension estimates for the set of points with non-dense orbit in homogeneous spaces

Dmitry Kleinbock, Shahriar Mirzadeh

math.DSmath.NT37A1737A2511J13

Paper metadata

Original submission: October 13, 2017
Latest revision: August 25, 2019 · version 2
Claimed MSC: 37A17, 37A25, 11J13
arXiv categories: math.DS, math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: 26 pages; more explanations added and several misprints corrected

Abstract

Let $X = G/Γ$ , where $G$ is a Lie group and $Γ$ is a lattice in $G$ , and let $U$ be a subset of $X$ whose complement is compact. We use the exponential mixing results for diagonalizable flows on $X$ to give upper estimates for the Hausdorff dimension of the set of points whose trajectories miss $U$ . This extends a recent result of Kadyrov and produces new applications to Diophantine approximation, such as an upper bound for the Hausdorff dimension of the set of weighted uniformly badly approximable systems of linear forms, generalizing an estimate due to Broderick and Kleinbock.

AI-generated audit

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

01Statements3 reported findingsCorrect

The dimension estimates for trajectories avoiding a co-bounded target, their compact-space corollary, the leafwise effective-equidistribution theorem, and the weighted badly approximable application are correct as stated.

Theorems 1.1 and 1.3Correct

Quantitative dimension drop for nondense trajectories

Pages 3–4 and 8–17 · Theorems 1.1, 1.3, and 4.1 · arXiv:1710.04898v2

Effective equidistribution gives a uniform positive proportion of each tessellation domain whose next translate enters the inner target. Iterating the resulting Bowen-box cover and converting boxes to metric balls yields the logarithmic codimension bound in Theorem 1.3. The stable/central and unstable local product decomposition then transfers this leafwise bound to $X$ , with the injectivity-radius and inner-core hypotheses transforming exactly as required for Theorem 1.1.

Corollary 1.2Correct

Neighborhood-avoidance estimate in the compact case

Pages 3 and 9–10 · Corollary 1.2 and its proof · arXiv:1710.04898v2

Applying Theorem 1.1 to $U=\partial_rS$ at scale $r/2$ gives a numerator containing $\mu(\partial_{r/2}S)$ . Compactness supplies a uniform ball-volume lower bound, and for an embedded $k$ -submanifold the tubular-neighborhood estimate gives $\mu(\partial_{r/2}S)\gg r^{\dim X-k}$ . These substitutions produce both displayed conclusions.

Theorem 1.4Correct

Weighted badly approximable systems have the announced codimension bound

Pages 4 and 22–25 · Theorem 1.4 and Section 8 · arXiv:1710.04898v2

Lemma 8.1 identifies $\operatorname{Bad}_{\mathbf i,\mathbf j}(c)$ with lattices avoiding the weighted cusp at radius $\varepsilon=c^{1/(m+n)}$ . The weighted cusp has measure comparable to $\varepsilon^{m+n}=c$ , and Proposition 8.3 shows that its inner core retains this order at a radius proportional to a fixed power of $\varepsilon$ . Substitution into Theorem 1.3 gives $\operatorname{codim}\operatorname{Bad}_{\mathbf i,\mathbf j}(c)\gg c/\log(1/c)$ with the asserted dependence of constants.

02Proofs6 reported findingsCorrect

The mixing-to-EEP argument, Bowen-box covering iteration, local-product reduction, and weighted-cusp estimates are correct and complete. Four printed notation, word, or symbol slips have uniquely determined local corrections and do not lower the proof status.

Sections 2–6Correct and complete

Effective equidistribution and the iterated covering argument

Pages 5–17 · Theorem 2.5, Theorem 4.1, and supporting lemmas · arXiv:1710.04898v2

Exponential mixing is smoothed at scale $e^{-\beta t}$ with the exponents balanced to produce EEP. Tessellation domains remove boundary losses when successive Bowen covers are multiplied. Proposition 5.1 supplies a uniform lower measure for the good portion, and Lemma 6.4 converts each final Bowen box into metric balls at radius $re^{-k\lambda_{\max}t}$ . The limiting covering exponent is precisely the claimed Hausdorff-dimension loss.

Lemma 7.2 and Section 7Correct and complete

Injectivity-radius control and weighted horospherical EEP

Pages 18–22 · Lemma 7.2 and Theorem 7.1 · arXiv:1710.04898v2

For successive-minimum vectors $v_1,\ldots,v_d$ , Minkowski's second theorem and the lattice determinant bound give the dual coefficient of $v_d$ norm $O(1/\lVert v_d\rVert)$ . The transvection fixing $v_1,\ldots,v_{d-1}$ and sending $v_d$ to $v_d+v_1$ is therefore $O(\delta(x)^{d/(d-1)})$ from the identity, proving the upper injectivity-radius bound. Quantitative nondivergence and the decomposition $g_t=a_tb_t$ then give the required uniform EEP estimate for the weighted subgroup.

Expansion-rate notationTypo

The contraction exponents are written as eigenvalues of $\operatorname{ad}_{g_1}$

Pages 10 and 15 · definitions of $\lambda_{\max}$ and $\lambda_0$ · arXiv:1710.04898v2

The element $g_1$ belongs to the group, so $\operatorname{ad}_{g_1}$ is not the infinitesimal adjoint map whose positive eigenvalues give the factors $e^{-\lambda t}$ used in the Bowen-box estimates. Let $D=\left.\frac{d}{dt}\right|_{t=0}g_t$ and replace these occurrences by $\operatorname{ad}(D)|_{\mathfrak p}$ ; equivalently, define the rates as the logarithms of the eigenvalues of $\operatorname{Ad}(g_1)|_{\mathfrak p}$ . The conjugation formulas immediately following the definitions already use exactly these rates, so this is a unique notation repair and no estimate changes.

Equation (3.6)Typo

The comparison constant is inverted

Page 9 · Equation (3.6) and the following parenthesis · arXiv:1710.04898v2

The proof sets $C'=2C$ when replacing $\log(4/r)+\log(1/\mu)$ by $\log(1/r)+\log(1/\mu)$ . Since $r<1/4$ , the first denominator is at most twice the second, so the needed lower-bound constant is $C'=C/2$ (or any smaller positive constant), and the displayed comparison sign in the parenthesis must be reversed accordingly. This is a local coefficient correction; the theorem requires only an unspecified positive uniform constant.

Proof of Corollary 1.2Typo

Codimension is bounded below, not above

Page 9 · two sentences following Equation (3.6) · arXiv:1710.04898v2

The prose twice says that the exceptional set has Hausdorff codimension at most the displayed positive quantity, while the formulas and the conclusion use a lower bound. Replace at most by at least in both sentences. No displayed inequality or subsequent deduction changes.

Proof of Theorem 4.1Typo

The final Bowen-ball radius repeats the iteration index

Page 17 · paragraph preceding Equation (6.8) · arXiv:1710.04898v2

The cover is once described as using balls of radius $re^{-k\lambda_{\max}k}$ . The conjugation time is $kt$ , and the preceding and following denominator formulas both use $re^{-k\lambda_{\max}t}$ . Replace the final $k$ in the exponent by $t$ ; the covering count and dimension limit are already computed with the corrected radius.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

Detailed audit reportFull reasoning, manuscript locations, and references.

Open report PDF ↗

Author response

Challenge an audit finding

Local workflow preview

A listed author may submit formal evidence that an audit is inaccurate. The response would be considered in a fresh AI re-evaluation; it would not edit the audit automatically.

Paper: arXiv:1710.04898v2
Authors listed: Dmitry Kleinbock, Shahriar Mirzadeh
Audit date: August 19, 2026

01Establish identityMatch an authenticated scholarly identity to this paper.
02Submit evidenceIdentify the finding and give a formal mathematical response.
03Re-evaluateA separate agent checks the response and records a disposition.

Email fallback for papers without a linked ORCID

A production fallback could send a one-time link only when the submitted address matches an independently maintained author-contact allowlist for this paper. MathAudit must return the same message for every address so the form cannot reveal which contacts are on that list.

Structured response preview

This form remains unavailable until production identity verification succeeds. Nothing entered here is submitted.

Audit finding challengedWhy the evaluation is inaccurateFormal evidence or repairVerifiable sources, if anyI am submitting evidence, not instructions for the reviewing agent.

This panel never establishes authorship in the local prototype. A production result should be described narrowly as an authenticated ORCID match or control of a separately allowlisted author-contact mailbox.

Dimension estimates for the set of points with non-dense orbit in homogeneous spaces

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Quantitative dimension drop for nondense trajectories

Neighborhood-avoidance estimate in the compact case

Weighted badly approximable systems have the announced codimension bound

Effective equidistribution and the iterated covering argument

Injectivity-radius control and weighted horospherical EEP

The contraction exponents are written as eigenvalues of ad⁡g1\operatorname{ad}_{g_1}adg1​​

The comparison constant is inverted

Codimension is bounded below, not above

The final Bowen-ball radius repeats the iteration index

Challenge an audit finding

Authenticate with ORCID, then require an exact arXiv match

The contraction exponents are written as eigenvalues of $\operatorname{ad}_{g_1}$