arXiv:2106.08860v2

Equidistribution in the space of 3-lattices and Dirichlet-improvable vectors on planar lines

Dmitry Kleinbock, Nicolas de Saxcé, Nimish A. Shah, Pengyu Yang

math.DSmath.NT37A1711J8322E4614L2411J13

Paper metadata

Original submission: June 16, 2021
Latest revision: August 30, 2022 · version 2
Claimed MSC: 37A17, 11J83, 22E46, 14L24, 11J13
arXiv categories: math.DS, math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: 35 pages, 2 figures

Abstract

Let $X=\text{SL}_3(\mathbb{R})/\text{SL}_3(\mathbb{Z})$ , and $g_t=\text{diag}(e^{2t}, e^{-t}, e^{-t})$ . Let $ν$ denote the push-forward of the normalized Lebesgue measure on a segment of a straight line in the expanding horosphere of $\{g_t\}_{t>0}$ , under the map $h\mapsto h\text{SL}_3(\mathbb{Z})$ from $\text{SL}_3(\mathbb{R})$ to $X$ . We give explicit necessary and sufficient Diophantine conditions on the line for equidistribution of each of the following families of measures on $X$ : (1) $g_t$ -translates of $ν$ as $t\to\infty$ . (2) averages of $g_t$ -translates of $ν$ over $t\in[0,T]$ as $T\to\infty$ . (3) $g_{t_i}$ -translates of $ν$ for some $t_i\to\infty$ . We apply this dynamical result to show that Lebesgue-almost every point on the planar line $y=ax+b$ is not Dirichlet-improvable if and only if $(a,b)\notin\mathbb{Q}^2$ .

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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

01Statements3 reported findingsCorrect

The criteria for nonescape, equidistribution, averaged equidistribution, and subsequential equidistribution of planar-line measures are correct, as is the almost-everywhere non-Dirichlet-improvability consequence. The reported definition-level notation errors have unique repairs and do not alter the results.

Theorems 1.1–1.4Correct

The Diophantine criteria exactly characterize the four limiting regimes

Pages 3–4 · Theorems 1.1–1.4 · arXiv:2106.08860v2

The standard-representation calculations identify bounded, vanishing, and positive-density short-vector obstructions with $W_2$ , $W_2'$ , and $W_2^+$ , respectively. Dani–Margulis nondivergence converts short vectors into escape of mass, while linearization, geometric invariant theory, and Ratner measure classification show that every nonequidistributing subsequence supplies exactly such a standard-representation obstruction. The rational case is separately characterized by a fixed vanishing vector, yielding the stated subsequential dichotomy.

Theorem 1.5Correct

Almost every point on an irrational planar line is not Dirichlet-improvable

Pages 5–6 · Theorem 1.5 and its deduction from Theorem 1.4 · arXiv:2106.08860v2

For each fixed $0<\delta<1$ , Dani's correspondence places every $\delta$ -improvable point in a nondense trajectory set. Theorem 1.4 makes each such set null when $(a,b)\notin\mathbb Q^2$ . Taking the countable union over $\delta=(m-1)/m$ , $m\geq2$ , covers every Dirichlet-improvable point, so the asserted almost-everywhere conclusion follows.

Definition preceding Theorem 1.5Typo

The fixed improvement parameter is accidentally requantified

Page 5 · definition of a $\delta$ -improvable linear form · arXiv:2106.08860v2

The paragraph begins with a fixed $0<\delta<1$ but then says the form is $\delta$ -improvable if there exists $0<\delta<1$ such that. Remove the repeated existential phrase, leaving the inequalities with the fixed displayed $\delta$ . This matches the immediately preceding vector definition and every subsequent use of $D_\delta$ .

02Proofs5 reported findingsCorrect

The nondivergence, linearization, representation-theoretic reduction, Ratner, and Diophantine arguments are correct and complete. Several literal index, variable, and quantifier slips have uniquely determined corrections and do not change the proof logic.

Sections 2–8Correct and complete

Linearization and the standard-representation obstruction

Pages 7–34 · Theorems 1.1–1.4 and their supporting propositions · arXiv:2106.08860v2

Kempf's instability criterion reduces every nonclosed orbit obstruction to a highest-weight vector and then to a fundamental representation. The exterior-square alternative is ruled out by a uniform lower bound, leaving a vector in $\mathbb Z^3$ . Explicit multiplication by $g_t\phi_{a,b}(s)$ gives the stated simultaneous approximation inequalities. In the closed-orbit case, the stabilizer analysis forces rationality. These alternatives cover loss of mass, failure of equidistribution, the averaged regime, and the dense-orbit subsequence without an omitted case.

Proof of Theorem 1.5Typo

The countable union starts with an undefined endpoint

Pages 5–6 · final sentence of the proof of Theorem 1.5 · arXiv:2106.08860v2

The proof prints $\bigcup_{m\geq1}D_{(m-1)/m}$ although $D_\delta$ was defined only for $0<\delta<1$ , so the first term would be $D_0$ . Replace $m\geq1$ by $m\geq2$ . Monotonicity in $\delta$ makes these rational parameters cofinal in $(0,1)$ , and the null-union argument is unchanged.

Proof of Theorem 1.1Typo

The sequence quantifier is reversed in the nonescape summary

Page 24 · proof of Theorem 1.1 · arXiv:2106.08860v2

The proof says that no escape of mass means there exists a sequence $t_i\to\infty$ for which Proposition 5.2(2) fails. It must say that the condition fails for every such sequence: a single sequence with no positive mass loss does not make the whole family tight. The surrounding argument and Lemma 3.2 already quantify over arbitrary sequences, so replacing there exists by for every is the unique correction and completes the stated implication.

Proposition 6.6Typo

Three variables in the averaged-measure argument are misprinted

Page 27 · Proposition 6.6 and its proof · arXiv:2106.08860v2

The average is printed as $\mu_i=T_i^{-1}\int_0^{T_i}g_{t_i}\lambda_{a,b}\,dt$ ; the integrand must be $g_t\lambda_{a,b}$ . Equation (6.4) prints $\liminf_{I\to\infty}$ instead of $\liminf_{i\to\infty}$ . In case (i), the upper bound is printed as $\mu_i(X)<1-\varepsilon$ , impossible for the probability measure $\mu_i$ ; it must be $\mu_i(K)<1-\varepsilon$ , as supplied by weak convergence on the chosen compact $K$ . These corrections are uniquely forced by the integration variable and the adjacent definitions, and the density estimate then follows.

Proof of Theorem 1.3Typo

The ambient lattice has the wrong dimension symbol

Page 33 · proof of Theorem 1.3, before Equation (8.12) · arXiv:2106.08860v2

The proof writes $g_t\phi_{a,b}(s)\mathbb Z^n\notin K$ , while throughout the paper $X$ is the space of unimodular $3$ -lattices. Replace $\mathbb Z^n$ by $\mathbb Z^3$ . The next displayed measure identity and Mahler argument already use the three-dimensional model.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2106.08860v2
Authors listed: Dmitry Kleinbock, Nicolas de Saxcé, Nimish A. Shah, Pengyu Yang
Audit date: August 19, 2026

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Equidistribution in the space of 3-lattices and Dirichlet-improvable vectors on planar lines

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The Diophantine criteria exactly characterize the four limiting regimes

Almost every point on an irrational planar line is not Dirichlet-improvable

The fixed improvement parameter is accidentally requantified

Linearization and the standard-representation obstruction

The countable union starts with an undefined endpoint

The sequence quantifier is reversed in the nonescape summary

Three variables in the averaged-measure argument are misprinted

The ambient lattice has the wrong dimension symbol

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