arXiv:2608.12420v1

A Kronecker-Weyl Theorem from Interpolation Sets

Rafael Reno S. Cantuba

math.FAmath.CO43A4611J71

Paper metadata

Original submission: August 12, 2026
Claimed MSC: 43A46, 11J71
arXiv categories: math.FA, math.CO
Paper license: CC BY 4.0 ↗

Abstract

An analog of the classical Kronecker-Weyl theorem for weak uniform distribution is obtained for an arbitrary countably infinite independent subset of a discrete abelian group using a type of interpolation sets called epsilon-Kronecker sets. Specifically, the topological size of the set of homomorphisms inducing weakly uniformly distributed sequences is established by showing that it forms a dense G-delta set.

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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.

Current report

Detailed mathematical audit

Generated August 15, 2026

01Statements2 reported findingsContains wrong statements

The advertised conclusion that $U(E,\mathbb T^{\mathbb T})$ is a dense $G_\delta$ set is false in the stated generality. The density-and-Baire conclusion printed as Theorem 5.1 is not able to be verified because its proof depends on two false lemmas and an invalid coordinate identification.

Advertised main resultIncorrect

The dense- $G_\delta$ conclusion is false

Page 1 · Abstract; page 3 · end of the introduction; page 9 · last sentence of the proof of Theorem 5.1 · arXiv:2608.12420v1

Take $\Gamma=\bigoplus_{n\geq1}\mathbb Z e_n$ and $E=\{e_n:n\geq1\}$ . These satisfy all stated hypotheses, and $X=\widehat\Gamma^{\mathbb T}$ is a compact product. Every $h\in U(E,\mathbb T^{\mathbb T})$ must have $h(s)\neq h(t)$ whenever $s\neq t$ : if $h(s)=h(t)$ , the nontrivial character $\chi_{s,t}(x)=x(s)x(t)^{-1}$ satisfies $\chi_{s,t}(\widetilde h(e_n))=1$ for every $n$ , contradicting weak uniform distribution already for $k=2$ . Now suppose $U$ were dense $G_\delta$ , say $U=\bigcap_r G_r$ with each $G_r$ open dense. Recursively choose nonempty basic cylinders $B_r$ whose closures are nested and satisfy $\overline{B_r}\subseteq B_{r-1}\cap G_r$ . These cylinders constrain only a countable set $J\subset\mathbb T$ of product coordinates. Choose distinct $s,t\notin J$ . Compactness gives a point in every $\overline{B_r}$ , and its unconstrained coordinates may be chosen with $h(s)=h(t)$ . This puts a noninjective $h$ in $\bigcap_rG_r=U$ , a contradiction. Thus the claimed dense- $G_\delta$ property is formally impossible.

Dikranjan–Shakhmatov, related warning that the Baire conclusion cannot generally be strengthened to dense $G_\delta$ ↗

Theorem 5.1Not able to verify

The printed density-and-Baire conclusion is not established

Page 9 · Theorem 5.1 and its proof · arXiv:2608.12420v1

The proof identifies the target set with the countable intersection in Lemma 3.1 and then invokes Lemma 4.3 to make its constituent open sets dense. The identification leading to Lemma 3.1 is false, the reduction from all characters to characters supported on a countable dense coordinate set is invalid, and Lemma 4.3's fixed- $N$ density assertion is false. These are necessary, nonlocal steps in the only supplied argument. They do not by themselves disprove the density or the Baire-subspace assertion stated in Theorem 5.1, but no verified proof of those remaining assertions is available from the paper.

Dikranjan–Shakhmatov, definition and use of the Baire-subspace property ↗

02Proofs4 reported findingsContains incorrect or incomplete proofs

The main proof contains several decisive formal failures: Equations (4)–(5) transpose unrelated coordinates, Lemma 3.1 uses continuity that points of a product group do not have, and Lemma 4.3 asserts density of sets that are not dense. The later reindexing also changes data fixed in the definition of those sets.

Equations (3)–(5)Incorrect as written

The claimed alternative characterization of $O_{N,\mu,k}$ is false

Pages 4–5 · Equations (3)–(5) · arXiv:2608.12420v1

For $h\in\widehat\Gamma^{\mathbb T}$ , let $\widetilde h(e)(t)=h(t)(e)$ , and write $g_m=\phi_\mu(\tau_m)$ . Correctly expanding the left side of (4) gives $\chi_\mu(h(t_n)\circ\phi_\mu)=\prod_m h(t_n)(g_m)^{n_m}=\prod_m\widetilde h(g_m)(t_n)^{n_m}.$ By contrast, the expression required in (5) is $\chi_\mu(\widetilde h(e_n))=\prod_m h(\tau_m)(e_n)^{n_m}.$ These use different elements and different product coordinates and are unrelated in general. The displayed intermediate term $h(t_n)(\tau_m)$ is not even well typed, since $h(t_n)$ has domain $\Gamma$ while $\tau_m\in\mathbb T$ . Consequently (5) and the ensuing Lemma 3.1 do not describe $U(E,\mathbb T^{\mathbb T})$ . Repair classification: No repair supplied; the defining open sets and the later interpolation construction would have to be rebuilt.

Full paper, version 1 ↗

Lemma 3.1Incorrect as written

A countable dense coordinate set cannot test all product-group characters by the stated argument

Pages 4–6 · Lemma 3.1, especially Equation (6) · arXiv:2608.12420v1

The proof declares $\Phi_x(s_1,\ldots,s_M)=\prod_m x(s_m)^{n_m}$ continuous as a function of $(s_1,\ldots,s_M)\in\mathbb T^M$ for every $x\in\mathbb T^{\mathbb T}$ . A point of the product $\mathbb T^{\mathbb T}$ is an arbitrary function $x:\mathbb T\to\mathbb T$ , not a continuous function, so this assertion is false. The claimed approximation of a character supported at arbitrary coordinates by characters supported on $\mathbb T_0$ therefore fails. Concretely, for the free group on $E$ , one may make every finite character supported on $\mathbb T_0$ have vanishing Cesàro averages while setting one coordinate $t_*\notin\mathbb T_0$ identically equal to $1$ along $E$ ; evaluation at $t_*$ then fails the weak-distribution test. Repair classification: No repair supplied; the uncountable family of product characters cannot be discarded through the asserted continuity.

Full paper, version 1 ↗

Lemma 4.3Incorrect as written

The fixed- $N$ sets $O_{N,\mu,k}$ are not dense

Pages 7–9 · Lemma 4.3 and proof of Theorem 5.1 · arXiv:2608.12420v1

At the identity point of $\widehat\Gamma^{\mathbb T}$ , every summand in the defining average for $O_{N,\mu,k}$ equals $1$ . For any $k\geq2$ , continuity of this finite average gives a nonempty open neighborhood on which its modulus remains greater than $1/k$ . That neighborhood is disjoint from $O_{N,\mu,k}$ , so the set is not dense. In the special case $N=1$ , the modulus is always $1$ , making $O_{1,\mu,k}$ empty for $k>1$ ; correspondingly, the proof's identity $N^{-1}\sum_{\nu=1}^N e^{2\pi i\nu/N}=0$ is false at $N=1$ . Repair classification: No repair supplied. A possible proof would need density of a union over suitably large $N$ , with $N$ chosen after the neighborhood constraints, rather than the false density of every fixed- $N$ set.

Full paper, version 1 ↗

Proof of Lemma 4.3Incomplete as written

The construction changes the enumeration and assignment map after the open set is fixed

Page 7 · Paragraph preceding Equation (11) · arXiv:2608.12420v1

The sets $O_{N,\mu,k}$ were defined using a fixed enumeration $(e_n)$ and fixed maps $\phi_\mu$ . Inside the proof, the manuscript reindexes $(e_n)$ to exclude a finite set $F$ and adjusts $\phi_\mu$ so that its values avoid $F$ . This replaces the first $N$ terms and the composition map in the finite average that defines the particular set $O_{N,\mu,k}$ ; it therefore changes the previously fixed set whose density is being proved. No invariance statement about these fixed- $N$ sets is supplied or possible under such a replacement. Repair classification: No repair supplied; the interpolation choices must be made without altering the fixed data, and their compatibility with an arbitrary basic neighborhood must then be proved.

Full paper, version 1 ↗

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2608.12420v1
Authors listed: Rafael Reno S. Cantuba
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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P(delta)

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P(delta)

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A Kronecker-Weyl Theorem from Interpolation Sets

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The dense-GδG_\deltaGδ​ conclusion is false

The printed density-and-Baire conclusion is not established

The claimed alternative characterization of ON,μ,kO_{N,\mu,k}ON,μ,k​ is false

A countable dense coordinate set cannot test all product-group characters by the stated argument

The fixed-NNN sets ON,μ,kO_{N,\mu,k}ON,μ,k​ are not dense

The construction changes the enumeration and assignment map after the open set is fixed

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The dense- $G_\delta$ conclusion is false

The claimed alternative characterization of $O_{N,\mu,k}$ is false

The fixed- $N$ sets $O_{N,\mu,k}$ are not dense