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Paper metadata

Original submission
March 25, 2024
Latest revision
November 25, 2025 · version 2
Claimed MSC
37A35, 28D20, 22D40
arXiv categories
math.DS
Paper license
CC BY 4.0 ↗
Comments
10 pages

Abstract

For any diagonal element aa with two eigenvalues, we construct a sequence of aa-invariant probability measures on the space of unimodular lattices with high entropy but converging to the zero measure. This extends the result of Kadyrov [Ergodic Theory Dynam. Systems, 32(1) (2012)].

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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 existence of invariant probability measures with entropy tending to m+n−1m+n-1 and total escape of mass, and the interpolated sharp escape-of-mass consequence, are correct.

Theorem 1.2Correct

High entropy with total escape of mass is established

Pages 2 and 8–9 · Theorem 1.2 · arXiv:2403.16840v2

Proposition 3.1 produces invariant probability measures supported where the shortest-vector function lies between two positive cutoffs that both tend to zero, while their entropies tend to m+n−1m+n-1. Mahler compactness forces every weak-star limit to be the zero measure, and the general entropy upper bound supplies the matching entropy limit.

Full paper, version 2 ↗
Corollary 1.3Correct

The sharp family of partial escape examples follows

Pages 2 and 9 · Corollary 1.3 · arXiv:2403.16840v2

Convex combinations with Haar measure preserve invariance, scale the escaping component by the prescribed mass, and make entropy affine. Substitution gives equality in the stated entropy-versus-mass bound.

02Proofs2 reported findingsCorrect

The variational and measure-construction arguments are correct. One reversed pair of bounds in the definition of an auxiliary set is a mechanically identifiable typo.

Proposition 3.1Typo

The bounds in the auxiliary set are reversed typographically

Page 6 · proof of Proposition 3.1, definition of E(ε)E(\varepsilon) · arXiv:2403.16840v2

The displayed definition prints ηε≤λ1(atuAZd)≤ρε\eta_{\varepsilon}\leq\lambda_1(a_tu_A\mathbb{Z}^d)\leq\rho_{\varepsilon}. Proposition 2.6, the preceding construction, and every subsequent use require ρε≤λ1(atuAZd)≤ηε\rho_{\varepsilon}\leq\lambda_1(a_tu_A\mathbb{Z}^d)\leq\eta_{\varepsilon}, with ρε<ηε\rho_{\varepsilon}<\eta_{\varepsilon}. Swapping the two endpoint symbols is the unique correction and leaves the argument unchanged.

Propositions 2.6 and 3.1Correct and complete

The variational construction has compatible support and entropy bounds

Pages 4–8 · Propositions 2.6 and 3.1 · arXiv:2403.16840v2

The dimension estimate supplies separated orbit segments inside a compact shortest-vector annulus. Averaging the resulting measures yields invariance, the entropy lower bound survives passage to a limit, and the support remains in the closed annulus.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:2403.16840v2
Authors listed
Taehyeong Kim
Audit date
August 19, 2026
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