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Metric mean dimension of amenable group actions: localization and non-uniformity

T0 review · 1 major / 2 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read The metric mean dimension of an amenable group action equals the asymptotic entropy of its pointwise ε-stable sets.

desk verdict Extends the localization formula to general amenable groups via the covering lemma and settles the uniformity questions with explicit counterexamples. read the letter →

arxiv 2606.13270 v1 pith:ZAHH7ESR submitted 2026-06-11 math.DS

Xinyao He , Guohua Zhang , Ruifeng Zhang This is my paper · ORCID
classification math.DS
keywords metricmeandimensionamenablegroupactionslocalizationformulapointwiseε-stablesetstopologicalentropypackingnon-uniformitycombinatorialcoveringlemma
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper extends Tsukamoto's localization formula for metric mean dimension from Z^k- and R^k-actions to actions of general countable discrete amenable groups. It establishes that the global metric mean dimension is recovered exactly from the asymptotic entropy of pointwise ε-stable sets. Equivalent definitions are given in terms of topological entropy, packing topological entropy, and Bowen's dimensional entropy. The proof replaces tiling arguments with Lindenstrauss's combinatorial covering lemma to accommodate the structure of arbitrary amenable groups. Counterexamples demonstrate that the supremum and limit superior in the localization formula cannot be interchanged, confirming non-uniform convergence.

What carries the argument

The localization formula that equates global metric mean dimension to the asymptotic entropy of pointwise ε-stable sets, which works by replacing tiling with Lindenstrauss's combinatorial covering lemma.

What would settle it

An amenable group action where the asymptotic entropy of pointwise ε-stable sets differs from the global metric mean dimension computed by other means would falsify the characterization.

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Extended reading notes

Core claim

The global metric mean dimension equals the asymptotic entropy of pointwise ε-stable sets for actions of countable discrete amenable groups. This is proved by introducing equivalent definitions via topological entropy, packing topological entropy, and Bowen's dimensional entropy, using Lindenstrauss's combinatorial covering lemma in place of tiling arguments, and constructing counterexamples that show the supremum and limit superior cannot be interchanged in the localization formula.

Load-bearing premise

Lindenstrauss's combinatorial covering lemma suffices to replace tiling arguments when defining the equivalent entropy notions for general amenable groups.

Editorial extensions

If this is right

  • Equivalent definitions via topological entropy, packing topological entropy, and Bowen's dimensional entropy provide flexible tools for computing metric mean dimension.
  • The supremum and limit superior in the localization formula cannot generally be interchanged.
  • The convergence appearing in the localization formula is heterogeneous.
  • The three uniformity questions from Yang, Chen, and Zhou are resolved by explicit counterexamples.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same covering-lemma technique may allow localization formulas for other entropy-like invariants on amenable groups.
  • Non-uniformity implies that pointwise dynamical features can dominate the value of global invariants even when averaged.
  • The new definitions could simplify explicit calculations of metric mean dimension on concrete amenable actions such as shifts on groups with complicated Følner sequences.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 2 minor

Summary. The paper extends Tsukamoto's localization formula for metric mean dimension from ℤ^k- and ℝ^k-actions to actions of countable discrete amenable groups. The central result (Theorem 2.3) asserts that the global metric mean dimension equals the asymptotic entropy of pointwise ε-stable sets. Equivalent definitions of the invariant are introduced via topological entropy, packing topological entropy, and Bowen's dimensional entropy. Tiling arguments are replaced by Lindenstrauss's combinatorial covering lemma to handle general amenable groups. Counterexamples (Theorem 2.6 and Proposition 5.8) are constructed to resolve the three uniformity questions from Yang-Chen-Zhou, showing that the supremum and limsup in the localization formula cannot be interchanged in general.

Significance. If the claims hold, the work generalizes an important invariant to a substantially larger class of group actions, supplies multiple equivalent characterizations that may aid concrete computations, and settles open uniformity questions via explicit counterexamples. The adaptation of the combinatorial covering lemma, if verified to apply without extra conditions, constitutes a useful technical contribution for amenable-group dynamics.

major comments (1)
  1. [Theorem 2.3] Theorem 2.3: The assertion that Lindenstrauss's combinatorial covering lemma suffices to establish equivalence among the topological, packing, and Bowen formulations (and thereby the localization formula) for arbitrary countable discrete amenable groups is load-bearing; the proof must explicitly confirm that the lemma applies directly to the relevant Følner sequences without additional restrictions on the group or the action.
minor comments (2)
  1. The definitions of pointwise ε-stable sets and the three entropy variants should be stated in full before their first use in the statements of the main theorems.
  2. Notation for the amenable group, the metric, and the Følner sequences should be introduced uniformly at the beginning of Section 2 to avoid later ambiguity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the positive evaluation of the manuscript. We address the single major comment below.

read point-by-point responses

  1. Referee: [Theorem 2.3] Theorem 2.3: The assertion that Lindenstrauss's combinatorial covering lemma suffices to establish equivalence among the topological, packing, and Bowen formulations (and thereby the localization formula) for arbitrary countable discrete amenable groups is load-bearing; the proof must explicitly confirm that the lemma applies directly to the relevant Følner sequences without additional restrictions on the group or the action.

    Authors: We agree that an explicit verification of the lemma's hypotheses is necessary for the argument to be fully rigorous. In the revised version we will add a short paragraph immediately after the invocation of Lindenstrauss's combinatorial covering lemma in the proof of Theorem 2.3. This paragraph will record that the Følner sequences employed are chosen to satisfy the exact conditions stated in Lindenstrauss's lemma (in particular, the vanishing of the boundary-to-volume ratio) and that the argument imposes no further restrictions on either the amenable group or the continuous action. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The derivation chain relies on external results including Tsukamoto's localization formula, Lindenstrauss's combinatorial covering lemma, and counterexamples resolving questions from Yang-Chen-Zhou. Equivalent definitions via topological, packing, and Bowen's entropies are introduced as technical tools without reducing to self-fitted parameters or self-citations. Theorem 2.3 characterizes global metric mean dimension via asymptotic entropy of pointwise ε-stable sets using these independent inputs, with no load-bearing self-citation chains or ansatzes smuggled from prior author work. The paper is self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The work rests on standard domain assumptions in ergodic theory for amenable groups and the applicability of Lindenstrauss's lemma; no free parameters or invented entities are introduced.

assumptions (1)
  • domain assumption Lindenstrauss's combinatorial covering lemma applies to countable discrete amenable groups and replaces tiling arguments
    Invoked to handle general amenable group structure in the proof of the localization formula.

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Cite this review

Pith. "Pith review of Metric mean dimension of amenable group actions: localization and non-uniformity." pith.science (2026). https://pith.science/paper/ZAHH7ESR

@misc{pith2026260613270,
  author       = {Pith},
  title        = {Pith review of: Metric mean dimension of amenable group actions: localization and non-uniformity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZAHH7ESR}},
  note         = {Machine review of arXiv:2606.13270}
}
abstract

In this paper, we extend Tsukamoto's recent localization formula for metric mean dimension to actions of countable discrete amenable groups, which previously applied only to $\mathbb{R}^k$- and $\mathbb{Z}^k$-actions -- by proving that the global metric mean dimension is characterized by the asymptotic entropy of pointwise $\varepsilon$-stable sets (Theorem 2.3). To achieve this generalization, we introduce equivalent definitions of the invariant using topological entropy, packing topological entropy, and Bowen's dimensional entropy, respectively. A key technical contribution is our replacement of tiling arguments with Lindenstrauss's combinatorial covering lemma, which enables us to handle the general structure of amenable groups. Furthermore, we resolve all three questions regarding uniformity raised in Section 6 of a recent paper by Yang, Chen, and Zhou by constructing counterexamples (Theorem 2.6 and Proposition 5.8), which demonstrates that the supremum and limit superior in the localization formula cannot generally be interchanged, thereby highlighting the heterogeneous nature of the convergence. These results clarify the uniformity issue and offer insights into the link between local dynamics and global invariants, while our equivalent definitions provide flexible tools for computing metric mean dimension in concrete settings.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Dimensional entropy of amenable group actions over stable sets and fibres

    math.DS 2026-06 unverdicted novelty 6.0 of 10

    Establishes characterizations of topological conditional entropy via dimensional entropy of stable sets and fibres for amenable group actions, with topological proofs generalizing prior work.

Reference graph

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