arXiv:2606.08199v1

Birkhoff genericity on affine subspaces in horospheres

Nimish A. Shah, Pengyu Yang

math.DSmath.NT37A1711J83

Paper metadata

Original submission: June 6, 2026
Claimed MSC: 37A17, 11J83
arXiv categories: math.DS, math.NT
Paper license: CC BY 4.0 ↗
Comments: 37 pages

Abstract

We study Birkhoff genericity for a simple uniformly expanding diagonal flow on $\mathrm{SL}_{n+1}(\mathbb R)/\mathrm{SL}_{n+1}(\mathbb Z)$ , with initial points restricted to affine subspaces of the expanding horospherical orbit through the identity coset. We prove that almost every point on such an affine subspace is Birkhoff generic, except possibly in two situations: either the defining matrix of the affine subspace has Diophantine exponent at least $n$ , or the affine subspace is arbitrarily well approximable by affine subspaces of dimension $(r-1)$ defined over a real number field of degree $m\ge 2$ , with $n+1=mr$ . As applications, we obtain Dirichlet non-improvability and logarithmic density results for almost every point on these affine subspaces.

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Audited against arXiv v1

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.

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Detailed mathematical audit

Generated August 18, 2026

01Statements4 reported findingsCorrect

The pointwise non-escape theorem, the equidistribution-versus-Liouville alternative, and the stated Diophantine consequences are supported under the strict hypothesis $\omega(A)<n$ .

Theorem 1.1Correct

Pointwise non-escape of mass

Pages 3 and 7–16 · Theorem 1.1 and Section 2 · arXiv:2606.08199v1

The Diophantine exponent bound gives a growth exponent strictly below the contraction threshold for the modified Margulis height. Proposition 2.7 then gives exponentially small measure for parameters whose trajectories spend too little time in a compact set. Borel–Cantelli and a countable sequence of error tolerances show that every weak-star limit has total mass one. The extension from integer to arbitrary averaging times follows by enlarging the compact set over a bounded time interval.

Theorem 1.2Correct

Equidistribution or a number-field Liouville obstruction

Pages 3–4 and 16–36 · Sections 3–4 · arXiv:2606.08199v1

Unipotent invariance and Ratner-type measure classification reduce a nonequidistributing limit to a singular homogeneous submanifold. Restriction of scalars supplies a real number field $F$ and an integer $r$ with $r[F:\mathbb Q]=n+1$ and $r\geq d+1$ . The second height function either avoids that singular set or yields rational Grassmannian approximants whose distance decays faster than every power of their height. The latter is exactly the stated $(F,r-1)$ -Liouville alternative.

Theorem 1.3Correct

Geometric formulation on projective subspaces

Pages 4–5 · Theorem 1.3 · arXiv:2606.08199v1

The quotient $P_a^-\backslash G$ is identified with projective space, and the finite measure is disintegrated over the affine charts of the prescribed projective subspace. Stable directions do not alter Birkhoff genericity, so Theorem 1.2 applies to almost every chart parameter. The exponent and number-field Liouville exclusions translate exactly to the projective definitions used in the statement.

Theorems 1.4–1.5Correct

Dirichlet non-improvability and logarithmic densities

Pages 5–6 · arXiv:2606.08199v1

The Dani correspondence turns eventual Dirichlet improvement into persistent avoidance of a positive-Haar-measure region, which is incompatible with Birkhoff genericity and proves the nullity assertion. Applying genericity to the cusp-indicator regions gives the logarithmic frequencies; approximation by continuity sets yields the stated continuous strictly increasing functions $f$ and $g$ .

02Proofs2 reported findingsCorrect

The two height-function arguments, homogeneous-measure reduction, and number-field approximation step form a complete proof. Several standard consequences are left to the reader, but they are immediate applications of the proved equidistribution theorem and require no correction.

Proposition 2.7Correct and complete

Uniform cusp control and non-escape

Pages 15–16 · arXiv:2606.08199v1

The modified Margulis height has a contraction exponent uniform on the parameter box because $\omega(A)<n$ makes the growth bound strict. Proposition 2.7 gives a summable exponential estimate for excessive cusp occupation. Borel–Cantelli, followed by a countable sequence of compact sets and tolerances, proves that every subsequential time-average retains full mass.

Proposition 4.4Correct and complete

From singular height to the Liouville obstruction

Pages 33–35 · arXiv:2606.08199v1

Inequalities (4.4)–(4.11) separate cusp height from singular height and produce perturbations $Y_i$ of size $e^{-(1-o(1))t_i}$ . Quantitative reduction theory gives representatives of height $e^{o(t_i)}$ , so the resulting projective subspaces $T_i$ approximate $A$ faster than every prescribed inverse power. This is exactly the $(F,r-1)$ -Liouville condition rather than merely an informal approximation statement.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2606.08199v1
Authors listed: Nimish A. Shah, Pengyu Yang
Audit date: August 18, 2026

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If the list is empty, the section may say only “No non-novelty findings.” PART 4 — TECHNICAL AND EVIDENTIARY RULES 1. Criticism must never be overstated. Call a statement incorrect, a proof incorrect or incomplete, or a result non-new only when strong formal evidence supports that exact judgment. Otherwise use Not able to verify where that outcome exists, or remain silent in the novelty section. 2. Review a specified paper version. Give exact page, theorem, equation, or proof locations; restate relevant quantifiers and hypotheses; and check domains, endpoints, parameter ranges, constants, and dependencies explicitly. Apply a defect only to the regimes and downstream results that actually depend on it. 3. Immediate-consequence closure: when deciding whether a conclusion is supported, treat the argument as including every immediate formal consequence of facts established earlier in the paper and available at that point. 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Birkhoff genericity on affine subspaces in horospheres

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Pointwise non-escape of mass

Equidistribution or a number-field Liouville obstruction

Geometric formulation on projective subspaces

Dirichlet non-improvability and logarithmic densities

Uniform cusp control and non-escape

From singular height to the Liouville obstruction

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