arXiv:2607.01655v1

Linking effective Ratner equidistribution to the semicircle law for skew-shift matrices

Cong Chen, Yong Li

math.DS37A1715B5237A3005C30

Paper metadata

Original submission: July 2, 2026
Claimed MSC: 37A17, 15B52, 37A30, 05C30
arXiv categories: math.DS
Paper license: arXiv non-exclusive distribution license ↗
Comments: 26 pages, 2 figures

Abstract

We consider large Hermitian matrices whose entries are defined by evaluating the exponential function along orbits of the skew-shift $\frac{j(j-1)}{2}ω+ jy + x \mod 1$ for irrational $ω$ . We establish a rigorous connection between the effective Ratner equidistribution theorem for unipotent orbits in $\mathrm{SL}(3,\mathbb{R})/\mathrm{SL}(3,\mathbb{Z})$ and the global semicircle law for such deterministic matrices. For frequency sequences satisfying a Diophantine condition, we prove that the empirical spectral distribution of these matrices converges to the Wigner semicircle law with optimal polynomial rate $O(N^{-1})$ ; for rectangular matrices the corresponding Marchenko--Pastur law is obtained. The proof uses a multi-parameter effective mixing property derived from the effective Ratner equidistribution theorem, combined with a graph-theoretic expansion of the moments. Our results evidence the quasirandom nature of the skew-shift dynamics observed in other contexts by Bourgain, Goldstein and Schlag, and Rudnick, Sarnak and Zaharescu, and provide a dynamical systems proof of the semicircle law with an improved convergence rate.

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Audit summary

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

01Statements2 reported findingsContains unsupported statements

The advertised semicircle-law result is not supported. Its Diophantine hypothesis is empty under the paper's own definition, and the central dynamical estimate omits parameter and Sobolev losses that are essential for any nonvacuous version.

Theorems 1.5 and 6.2Not able to verify

The stated Diophantine range contains no irrational number

Pages 5–6 and 23–24 · Corollary 1.2, Theorems 1.5 and 6.2 · arXiv:2607.01655v1

The paper defines exponent $\kappa$ by $\lVert q\alpha\rVert\geq c q^{-\kappa}$ for every $q\geq1$ , then assumes $\kappa\leq0.6$ . For every irrational $\alpha$ , continued-fraction convergents satisfy $\lVert q\alpha\rVert<1/q$ infinitely often. If $\kappa<1$ , the ratio $(1/q)/q^{-\kappa}=q^{\kappa-1}$ tends to zero, so no positive $c$ can satisfy the definition. Thus the theorem is formally vacuous and cannot establish the claimed result for a concrete frequency.

Full paper, version 1 ↗

Lemma 6.1 and abstractNot able to verify

The claimed absolute decay rate does not follow from the displayed estimate

Abstract and pages 22–24 · equations (14)–(15), Lemma 6.1 · arXiv:2607.01655v1

The effective equidistribution theorem quoted on page 5 bounds the error by $e^{-\eta t}\lVert F\rVert_S$ . In Section 6, the test function $\Phi$ depends on $N$ and the paper itself records $\lVert\Phi\rVert_S\ll N^{C_0}$ , but equation (15) replaces the error by $e^{-\eta t}=N^{-6\eta}$ and drops the factor $N^{C_0}$ . The asserted constant also cannot be uniform in $1\leq s\leq N$ : the Diophantine constant of $s\alpha$ deteriorates with $s$ . Hence neither Lemma 6.1 nor the abstract's stronger 'optimal $O(N^{-1})$ ' claim is established.

02Proofs2 reported findingsContains incorrect or incomplete proofs

The main proof contains multiple nonrepairable-as-written gaps: it applies a continuous homogeneous-space equidistribution estimate to a discrete, $N$ -dependent torus observable without a proved identification and discards its Sobolev loss.

Sections 3 and 6.3Incorrect or incomplete

The asserted lift does not justify the discrete average

Pages 16–18 and 23 · Lemma 3.1 and Section 6.3 · arXiv:2607.01655v1

Section 6 says that a torus function can be lifted to an automorphic function so that a finite Birkhoff sum $N^{-1}\sum_{m=1}^N|\Phi(s\alpha m)|^2$ equals evaluation along an expanding unipotent curve, but no map with this identity is defined or proved. Yang's quoted theorem controls an integral over a continuous curve from a good base point, not this discrete sum. The inductive multi-parameter lemma likewise assumes goodness for shifted base points without verifying the alternative in the quoted theorem.

Equations (14)–(15)Incorrect

The smoothing and effective-error bookkeeping is invalid

Pages 22–23 · equations (14)–(15) · arXiv:2607.01655v1

With $Q=N^2$ , the stated truncation error $O(Q/N)$ is $O(N)$ , so after the displayed normalization it is not the claimed $O(N^{-1})$ . More decisively, the effective estimate must carry the growing Sobolev norm of the $N$ -scale bump. Dropping that factor changes a potentially growing error into a decaying one and invalidates the melon-sum bound on which all moment estimates depend.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2607.01655v1
Authors listed: Cong Chen, Yong Li
Audit date: August 18, 2026

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Linking effective Ratner equidistribution to the semicircle law for skew-shift matrices

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The stated Diophantine range contains no irrational number

The claimed absolute decay rate does not follow from the displayed estimate

The asserted lift does not justify the discrete average

The smoothing and effective-error bookkeeping is invalid

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