arXiv:2207.13155v2

Dimension drop for diagonalizable flows on homogeneous spaces

Dmitry Kleinbock, Shahriar Mirzadeh

math.DSmath.NT37A1737A2511J13

Paper metadata

Original submission: July 26, 2022
Latest revision: August 5, 2022 · version 2
Claimed MSC: 37A17, 37A25, 11J13
arXiv categories: math.DS, math.NT
Paper license: CC BY 4.0 ↗
Comments: 33 pages; a reference and acknowledgement added

Abstract

Let $X = G/Γ$ , where $G$ is a Lie group and $Γ$ is a lattice in $G$ , let $O$ be an open subset of $X$ , and let $F = \{g_t: t\ge 0\}$ be a one-parameter subsemigroup of $G$ . Consider the set of points in $X$ whose $F$ -orbit misses $O$ ; it has measure zero if the flow is ergodic. It has been conjectured that this set has Hausdorff dimension strictly smaller than the dimension of $X$ . This conjecture is proved when $X$ is compact or when $G$ is a simple Lie group of real rank $1$ , or, most recently, for certain special flows on the space of lattices. In this paper we prove this conjecture for arbitrary $\operatorname{Ad}$ -diagonalizable flows on irreducible quotients of semisimple Lie groups. The proof uses exponential mixing of the flow together with the method of integral inequalities for height functions on $G/Γ$ . We also derive an application to jointly Dirichlet-Improvable systems of linear forms.

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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

01Statements4 reported findingsCorrect

The abstract dimension-drop theorem under effective equidistribution and effective nondivergence, its semisimple homogeneous-space corollary, and the jointly Dirichlet-improvable dimension bound are correct. The matrix-space symbols in the last application have a unique harmless correction.

Theorem 1.5Correct

EEP and ENDP imply uniform leafwise dimension drop

Pages 4–7 and 26–29 · Theorems 1.5, 2.1, and Section 9 · arXiv:2207.13155v2

EEP supplies a uniform covering loss for orbit segments remaining in a suitable compact set, while ENDP supplies an independent covering loss for segments making repeated cusp excursions. Proposition 8.1 combines the two alternatives over blocks, and Theorem 2.1 converts the resulting covering number into positive Hausdorff codimension. The final choices of the compact set, scale, block length, and Margulis constant make the lower bound positive uniformly in the base point.

Corollary 1.6Correct

Dimension drop on the full homogeneous space

Pages 5 and 7–8 · Corollary 1.6 and its proof · arXiv:2207.13155v2

For the semisimple products in Theorem 1.4, the expanding horospherical subgroup has ENDP from the cited height-function construction and EEP from exponential mixing. The local product of expanding and nonexpanding directions lets the leafwise bound of Theorem 1.5 be sliced with the complementary factor, giving dimension strictly below $\dim X$ for the global exceptional set.

Theorem 10.1Correct

Joint Dirichlet improvement has dimension drop

Pages 30–31 · Theorem 10.1 and Equations (10.3)–(10.5) · arXiv:2207.13155v2

The lattice correspondence turns joint $c$ -improvement into eventual avoidance of $O^k$ by the product diagonal flow. The product expanding horospherical subgroup has ENDP componentwise and EEP because exponential mixing tensorizes over finitely many factors. Theorem 1.5 therefore gives positive codimension in the $kmn$ -dimensional matrix parameter space.

Definition preceding Theorem 10.1Typo

The matrix dimensions are transposed

Page 31 · definition of jointly $c$ -Dirichlet-improvable tuples and Equation (10.4) · arXiv:2207.13155v2

The text places $(Y_1,\ldots,Y_k)$ in $M_{n,m}^k$ , but $Y_iq-p$ with $q\in\mathbb Z^n$ and $p\in\mathbb Z^m$ requires $Y_i\in M_{m,n}$ . Replace both displayed occurrences of $M_{n,m}^k$ by $M_{m,n}^k$ . Also replace the mixed delimiter $|Y_iq-p\|$ in (10.4) by $\|Y_iq-p\|$ . The earlier definition, the matrices $h_Y$ , and the dimension $kmn$ make these corrections unique.

02Proofs6 reported findingsCorrect

The EEP covering, Margulis-inequality iteration, combined block argument, and product-flow application are correct and complete. Four literal normalization, rate-notation, target-letter, and property-name errors, together with one local inner-core scale correction, have unique repairs and do not change the arguments.

Sections 4–9Correct and complete

Compact and cuspidal covering estimates combine correctly

Pages 11–29 · Propositions 4.4, 6.1, 8.1, and proof of Theorem 2.1 · arXiv:2207.13155v2

The smoothed indicator estimate from EEP bounds the number of compact-part Bowen boxes. Iterating the Margulis inequality controls the height integral over cusp blocks, and Markov's inequality gives the ENDP covering count. Proposition 8.1 partitions itineraries by the two block types and keeps the overlap factors explicit. The parameter choices in Section 9 make the combined exponential base smaller than the full Bowen growth rate, which yields the claimed codimension.

Haar-measure conventionTypo

The normalized ball radius should be one

Page 2 · notation paragraph before Definition 1.1 · arXiv:2207.13155v2

The paper says that Haar measure is normalized so that $\nu(B^P(r))=1$ , which cannot hold for every variable radius $r$ . Replace $r$ by $1$ . Section 6 explicitly uses $\nu(B^P(1))=1$ , confirming the unique intended normalization.

Expansion-rate notationTypo

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

Pages 5–6 and 9 · Equations (2.3), (2.4), and the definition of $\delta$ · arXiv:2207.13155v2

The element $g_1$ belongs to the group, so $\operatorname{ad}_{g_1}$ is not the infinitesimal adjoint map whose positive eigenvalues produce the factors $e^{-\lambda t}$ and the Haar scaling $e^{-\delta t}$ . Let $D=\left.\frac{d}{dt}\right|_{t=0}g_t$ and use $\operatorname{ad}(D)|_{\mathfrak p}$ in these definitions; equivalently, take logarithms of the eigenvalues of $\operatorname{Ad}(g_1)|_{\mathfrak p}$ . All following conjugation and measure formulas already use these infinitesimal rates, making the repair unique and harmless.

Proof of Theorem 1.5Minor formal correction

The inner-core scale must be chosen below the defining supremum

Pages 6–7 · definition of $\theta_O$ and Equation (2.14) in the proof of Theorem 1.5 · arXiv:2207.13155v2

Equation (2.14) must read $\mu(\sigma_{4\theta}O)\geq\mu(O)/2$ , not $\leq\mu(O)/2$ , because the next step subtracts two error terms from this quantity. In addition, the printed choice $\theta=\min(\theta_O,r_*/2)$ can attain the supremum in the definition of $\theta_O$ , while the definition alone guarantees the measure inequality only strictly below that supremum. Replace it by $\theta=\min(\theta_O/2,r_*/2)$ . Since the later radius satisfies $r\leq\theta_O/4$ , one still has $r\leq\theta\leq r_*/2$ ; all subsequent parameter conditions and the codimension bound are unchanged. This local choice and the sign correction give the required lower bound $\mu(O)/4$ without assuming an unstated continuity property.

Proof of Corollary 1.6Typo

The target set changes from $O$ to an undefined $U$

Page 8 · paragraph before the Wegmann product estimate · arXiv:2207.13155v2

After fixing the open set $O$ , the proof says that $hx$ belongs to $E(F,\sigma_{2\rho}U)$ . Replace $U$ by $O$ . The preceding distance estimate and every set in the following dimension calculation use $\sigma_{2\rho}O$ .

Proof of Theorem 10.1Typo

Exponential mixing supplies EEP, not a second copy of ENDP

Page 31 · paragraph immediately before Theorem 10.1 · arXiv:2207.13155v2

Theorem 1.4 first supplies ENDP for $H^{(k)}$ . The next sentence invokes exponential mixing and Theorem 1.2 but again says that $H^{(k)}$ has ENDP. Replace the second ENDP by EEP. With both distinct properties available, the stated application of Theorem 1.5 follows.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2207.13155v2
Authors listed: Dmitry Kleinbock, Shahriar Mirzadeh
Audit date: August 19, 2026

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Dimension drop for diagonalizable flows on homogeneous spaces

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

EEP and ENDP imply uniform leafwise dimension drop

Dimension drop on the full homogeneous space

Joint Dirichlet improvement has dimension drop

The matrix dimensions are transposed

Compact and cuspidal covering estimates combine correctly

The normalized ball radius should be one

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

The inner-core scale must be chosen below the defining supremum

The target set changes from OOO to an undefined UUU

Exponential mixing supplies EEP, not a second copy of ENDP

Challenge an audit finding

Authenticate with ORCID, then require an exact arXiv match

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

The target set changes from $O$ to an undefined $U$