arXiv:2605.15580v2

A Weyl-type theorem for Diophantine approximations driven by LCA groups and applications

Aihua Fan

math.DSmath.CAmath.NT37B0511J1343A07

Paper metadata

Original submission: May 15, 2026
Latest revision: June 26, 2026 · version 2
Claimed MSC: 37B05, 11J13, 43A07
arXiv categories: math.DS, math.CA, math.NT
Paper license: CC BY-NC-SA 4.0 ↗

Abstract

We investigate actions of locally compact Abelian (LCA) groups on the torus $\mathbb{T}^n$ , motivated by their close connection with Diophantine approximation. While Kronecker's theorem yields a classical density result, we prove a stronger equidistribution theorem of Weyl type: every such action admits a decomposition into uniquely ergodic subsystems. The proof of this result is based on a characterization of unique ergodicity for actions of amenable groups on compact metric spaces. As consequences, we establish several foundational results for LCA groups, including the Bohr orthogonality of characters along arbitrary Folner sequences, a Bohr mean formula for almost periodic functions, and a Wiener-type theorem on LCA groups characterizing the discrete part of a Borel probability measure through its Fourier transform. An application to numerical analysis is also discussed.

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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 15, 2026

01Statements4 reported findingsCorrect

The central statements checked are correct. The orbit decomposition into minimal uniquely ergodic cosets, the affine classification of conjugacies, Bohr orthogonality, the Wiener formulas, the Følner formula for the Bohr mean, and the projection-flow criterion all follow under their stated hypotheses. The notation defects listed in Part 2 are mechanically repairable and do not change any result.

Theorems 1.1–1.2Correct

Kronecker criterion and uniquely ergodic coset decomposition

Pages 1–3 and 8–13 · Theorems 1.1–1.2 and Sections 2–3.2 · arXiv:2605.15580v2

The annihilator calculation identifies the orbit closure $H$ with the subgroup determined by all integer relations among $g_1,\ldots,g_n$ . Translation by the dense subgroup forces every invariant probability measure on $H$ to be Haar measure, and every orbit in a coset $zH$ is dense. The Følner unique-ergodicity criterion then yields the asserted uniform equidistribution, while $H=\mathbb T^n$ is equivalent to the stated absence of a nonzero integer relation.

Full paper, version 2 ↗

Theorem 1.3Correct

Affine classification of conjugacies between minimal translation actions

Pages 4 and 13–15 · Theorem 1.3 and Section 3.4 · arXiv:2605.15580v2

After normalizing a conjugacy to fix the identity, the cocycle $B(z,w)=K(zw)K(z)^{-1}K(w)^{-1}$ is continuous and invariant in $z$ . Minimality makes it constant, so $K$ is a continuous torus automorphism and hence is induced by a matrix in $\mathrm{GL}(n,\mathbb Z)$ . Evaluating the conjugacy on the translation orbit gives $h=Pg$ , and minimality eliminates every row of the difference of two possible matrices, proving uniqueness.

Full paper, version 2 ↗

Theorems 1.4–1.6Correct

Bohr orthogonality, Wiener formulas, and the Bohr mean

Pages 4–6 and 19–21 · Theorems 1.4–1.6 and Section 4 · arXiv:2605.15580v2

For a nontrivial character, either unique ergodicity on its compact orbit or the direct translated-Følner identity makes the normalized average vanish. Dominated convergence and Fubini then recover each atom and the squared mass of the discrete part of a finite measure. Uniform approximation by trigonometric polynomials proves existence, translation-uniformity, and Følner-sequence independence of the mean for every Bohr almost periodic function.

Full paper, version 2 ↗

Theorem 5.1Correct

Unique ergodicity criterion for the projected flow

Pages 21–24 · Section 5 and Theorem 5.1 · arXiv:2605.15580v2

The projected flow is a translation flow on a torus. Computing the annihilator of its orbit closure shows that the closure is the whole torus exactly when the only integer relation among the displayed frequencies is the zero relation. Theorem 1.2 then gives the claimed equivalence with unique ergodicity.

Full paper, version 2 ↗

02Proofs4 reported findingsCorrect

The proofs of the central statements and their material inputs are correct and complete. The closure argument in the amenable unique-ergodicity criterion, the Fourier proof on each coset, and the applications in harmonic analysis all preserve the required quantifiers. Four literal index or ambient-space mismatches are reported in yellow and do not lower the proof status.

Theorems 2.1–2.2Correct and complete

Følner characterization of unique ergodicity

Pages 8–10 · Section 2 · arXiv:2605.15580v2

Coboundary averages vanish uniformly by the Følner property. The Hahn–Banach argument identifies the kernel of the limiting projection with the closed span of coboundaries, and weak-star limits of empirical measures prove that unique ergodicity forces a point-independent limit. Approximation of continuity-set indicators by continuous functions gives the stated Riemann-integrable extension.

Full paper, version 2 ↗

Theorems 1.1–1.3Correct and complete

Annihilator, Haar-measure, and conjugacy arguments

Pages 11–15 · Sections 3.1–3.4 · arXiv:2605.15580v2

Pontryagin duality identifies the annihilator exactly, Fourier coefficients characterize Haar measure on the orbit closure, and translation by a coset representative transfers the result to every subsystem. The normalized-conjugacy argument uses only continuity and minimality before invoking the standard classification of continuous automorphisms of $\mathbb T^n$ .

Full paper, version 2 ↗

Lemmas 3.1–3.2Typo

Two Fourier-coordinate exponents repeat the first index

Page 11 · Proofs of Lemmas 3.1 and 3.2 · arXiv:2605.15580v2

In Lemma 3.1, replace the last exponent $u_1$ in $\gamma(g_1)^{u_1}\cdots\gamma(g_n)^{u_1}$ by $u_n$ . In Lemma 3.2, replace the last factor $\gamma(g_1)$ in the coordinate product by $\gamma(g_n)$ . The definitions of $\chi_u$ and $\Phi_\gamma$ uniquely force both corrections, and the displayed conclusions already use $\gamma(u_1g_1+\cdots+u_ng_n)$ .

Full paper, version 2 ↗

Proofs of Theorems 2.1 and 1.2(d)Typo

A subsequence label and the ambient torus are misnamed

Pages 10 and 12 · Part A of Section 2.2 and proof of Theorem 1.2(d) · arXiv:2605.15580v2

In the two-subsequence contradiction, the second limit must read $A_{n_j''}\varphi(z_0)\to b$ , not a second occurrence of $A_{n_j'}$ . In the proof of Theorem 1.2(d), replace $H=G$ by $H=\mathbb T^n$ : $H$ is a subgroup of the torus, and unique ergodicity of the action on the whole torus is equivalent to its orbit closure being the whole torus. The surrounding definitions uniquely determine both repairs.

Full paper, version 2 ↗

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2605.15580v2
Authors listed: Aihua Fan
Audit date: August 15, 2026

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You are a mathematical auditor. Evaluate only the correctness of the paper's central statements, the correctness and completeness of its proofs, and evidence-backed non-novelty. Do not make editorial comments or evaluate importance, exposition, style, or suitability for publication. Keep the report neutral, precise, and professional. PART 1 — CORRECTNESS OF THE MAIN STATEMENTS Identify the paper's central statements and assess each one separately using exactly one of these outcomes: - Correct: the statement has a correct, verifiable proof; or, despite a defect in the printed proof, you can give a fully verified repair or independent proof. - Incorrect: you can give strong formal evidence that the statement as written is false, preferably a counterexample or a complete contradiction. - Not able to verify: the supplied proof does not establish the statement, and you can neither supply a fully verified proof nor disprove it. 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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. An immediate formal consequence is a local, mechanically checkable inference using only definitions already in force, elementary logic, direct substitution or renaming, weakening of an inequality or set inclusion, or a monotonicity or symmetry property explicit in a definition or already proved. 4. Such an inference must introduce no new lemma, estimate, construction, case analysis, limiting or compactness argument, external result, or new issue of uniformity or parameter dependence. Check the direction of monotonicity and the dependence of every threshold or constant explicitly. Do not exchange pointwise and uniform conclusions, change exceptional sets, interchange quantifiers or limits, or extend an endpoint under this rule unless the required fact was already established. 5. If the claimed conclusion follows after immediate-consequence closure, it is fully supported. 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P(delta)

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P(delta)

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A Weyl-type theorem for Diophantine approximations driven by LCA groups and applications

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Kronecker criterion and uniquely ergodic coset decomposition

Affine classification of conjugacies between minimal translation actions

Bohr orthogonality, Wiener formulas, and the Bohr mean

Unique ergodicity criterion for the projected flow

Følner characterization of unique ergodicity

Annihilator, Haar-measure, and conjugacy arguments

Two Fourier-coordinate exponents repeat the first index

A subsequence label and the ambient torus are misnamed

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