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arxiv: 2605.15580 · v1 · pith:XDNXJSRAnew · submitted 2026-05-15 · 🧮 math.DS · math.CA· math.NT

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

Aihua Fan This is my paper

Pith reviewed 2026-05-19 19:55 UTC · model grok-4.3

classification 🧮 math.DS math.CAmath.NT
keywords locally compact abelian groupsunique ergodicityWeyl equidistributionDiophantine approximationBohr orthogonalityFølner sequencesalmost periodic functions
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The pith

Every action of a locally compact Abelian group on the torus decomposes into uniquely ergodic subsystems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that any action of a locally compact Abelian group on the n-torus breaks into subsystems that are each uniquely ergodic. This turns Kronecker's density result into a full equidistribution statement of Weyl type. The argument rests on an existing characterization of unique ergodicity that holds for amenable groups acting on compact metric spaces. A reader would care because the decomposition supplies concrete tools for Diophantine problems, such as orthogonality of characters along Følner sequences and mean-value formulas for almost periodic functions.

Core claim

Every action of a locally compact Abelian group on the torus admits a decomposition into uniquely ergodic subsystems. This decomposition produces a Weyl-type equidistribution theorem for the action.

What carries the argument

The characterization of unique ergodicity for amenable-group actions on compact metric spaces, which directly yields the decomposition of the given LCA actions.

If this is right

  • Bohr orthogonality holds for characters of LCA groups along any Følner sequence.
  • A mean-value formula holds for almost periodic functions on LCA groups.
  • A Wiener-type theorem characterizes the discrete part of any Borel probability measure on an LCA group by the behavior of its Fourier transform.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same decomposition technique may be tried on concrete low-dimensional cases such as the real line acting on the circle to produce explicit equidistribution rates.
  • The result suggests that similar unique-ergodicity splittings could be sought for actions on other compact groups beyond the torus.

Load-bearing premise

The pre-existing characterization of unique ergodicity for amenable group actions on compact metric spaces applies to the specific LCA actions on the torus.

What would settle it

An explicit LCA-group action on the torus that cannot be decomposed into any collection of uniquely ergodic subsystems would refute the central theorem.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript proves a Weyl-type equidistribution theorem for continuous actions of locally compact Abelian (LCA) groups on the n-torus: every such action decomposes into uniquely ergodic subsystems. From this it derives Bohr orthogonality of characters along arbitrary Følner sequences, a Bohr mean formula for almost periodic functions, and a Wiener-type theorem characterizing the discrete part of a Borel probability measure on an LCA group via its Fourier transform. The proof invokes a pre-existing characterization of unique ergodicity for amenable group actions on compact metric spaces.

Significance. If the decomposition holds, the result strengthens Kronecker's density theorem to a genuine equidistribution statement in the Diophantine-approximation setting and supplies useful tools for harmonic analysis on LCA groups. The approach correctly exploits amenability of LCA groups; the listed consequences are natural and potentially applicable once the central step is secured.

major comments (1)
  1. [Abstract and proof of the main decomposition theorem] Abstract (description of the proof): the decomposition into uniquely ergodic subsystems is obtained by direct appeal to a characterization of unique ergodicity for amenable actions on compact metric spaces. The manuscript does not verify that the hypotheses of this characterization (sigma-compactness, existence of a Følner net compatible with the uniform structure of the action, or any required continuity conditions on the torus metric) hold for arbitrary LCA groups, including non-discrete or non-second-countable cases, acting continuously on T^n. Because the subsequent Bohr orthogonality and Wiener-type results rest on this decomposition, the gap is load-bearing for the central claim.
minor comments (2)
  1. The abstract refers to 'arbitrary Følner sequences' for the orthogonality result; it would be useful to state explicitly whether the argument requires a specific net or works for every Følner net.
  2. Notation for the torus action and the induced Følner averages should be introduced with a short paragraph before the statement of the main theorem to improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on the manuscript. The primary concern is the need for explicit verification that the hypotheses of the invoked characterization of unique ergodicity hold for arbitrary LCA groups acting continuously on T^n. We address this point below and will revise the manuscript accordingly to strengthen the proof.

read point-by-point responses

  1. Referee: [Abstract and proof of the main decomposition theorem] Abstract (description of the proof): the decomposition into uniquely ergodic subsystems is obtained by direct appeal to a characterization of unique ergodicity for amenable actions on compact metric spaces. The manuscript does not verify that the hypotheses of this characterization (sigma-compactness, existence of a Følner net compatible with the uniform structure of the action, or any required continuity conditions on the torus metric) hold for arbitrary LCA groups, including non-discrete or non-second-countable cases, acting continuously on T^n. Because the subsequent Bohr orthogonality and Wiener-type results rest on this decomposition, the gap is load-bearing for the central claim.

    Authors: We agree that the manuscript would benefit from an explicit verification of the hypotheses of the cited characterization of unique ergodicity for amenable actions on compact metric spaces. In the revised version, we will insert a dedicated remark or short subsection immediately after the statement of the main decomposition theorem. There we will confirm that every LCA group is amenable and admits Følner nets (or nets in the general case), that the continuous action on the compact metric space T^n is compatible with the uniform structure, and that the relevant continuity conditions on the torus metric are satisfied. For non-second-countable LCA groups we will note that the relevant orbit closures remain compact metric spaces, so the decomposition into uniquely ergodic subsystems continues to hold. This addition will directly support the subsequent derivations of Bohr orthogonality and the Wiener-type theorem. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external characterization

full rationale

The paper derives its Weyl-type equidistribution theorem by invoking a pre-existing characterization of unique ergodicity for amenable group actions on compact metric spaces and applying it to LCA group actions on the torus. This is an external result rather than a self-referential definition, fitted input, or self-citation chain that reduces the conclusion to the inputs by construction. No equations or steps in the abstract or described proof chain rename known results, smuggle ansatzes, or force predictions from subsets of the same data. The argument remains self-contained against external benchmarks, with the decomposition and subsequent Bohr orthogonality and Wiener-type results following from the application of the cited characterization without definitional loops.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of a known characterization of unique ergodicity for amenable actions; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Characterization of unique ergodicity for actions of amenable groups on compact metric spaces
    Invoked as the basis for proving the decomposition of LCA group actions.

pith-pipeline@v0.9.0 · 5651 in / 1237 out tokens · 35557 ms · 2026-05-19T19:55:00.275916+00:00 · methodology

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Reference graph

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