arXiv:2412.02278v3

Mean dimension theory for infinite dimensional Bedford-McMullen sponges

Qiang Huo

math.DS28A8037C4528D2037B4037A35

Paper metadata

Original submission: December 3, 2024
Latest revision: August 16, 2026 · version 3
Claimed MSC: 28A80, 37C45, 28D20, 37B40, 37A35
arXiv categories: math.DS
Paper license: arXiv non-exclusive distribution license ↗
Comments: V2: Accepted for publication in Discrete and Continuous Dynamical Systems

Abstract

Tsukamoto (2022) introduced the notion of Bedford-McMullen carpet system, a subsystem of $([0,1]^{\mathbb{N}}\times[0,1]^{\mathbb{N}},shift)$ whose metric mean dimension and mean Hausdorff dimension does not coincide in general. The aim of this paper is to develop the mean dimension theory for Bedford-McMullen sponge system, which is a subsystem of $(([0,1]^r)^{\mathbb{N}},shift)$ with arbitrary $3\leq r\in\mathbb{N}$ . In particular, we compute the metric mean dimension and mean Hausdorff dimension of such topological dynamical systems explicitly, extending the results by Tsukamoto. The metric mean dimension is a weighted combination of the standard topological entropy, whereas the mean Hausdorff dimension is expressed in terms of weighted topological entropy. We also exhibit a special situation for which the metric mean dimension and the mean Hausdorff dimension of a sponge system coincide.

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Audited against arXiv v3

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

01Statements3 reported findingsCorrect

The two formulas in Theorem 1.1 correctly compute the metric mean dimension and mean Hausdorff dimension of the stated infinite-dimensional Bedford-McMullen sponge systems. The special uniform-fibre consequence in Lemma 3.4 is also correct.

Theorem 1.1, formula (3)Correct

Metric mean dimension is the stated entropy combination

Pages 3-4 and 7-10 · Theorem 1.1 and Section 3.1 · arXiv:2412.02278v3

At scale $m_1^{-M}$ , the first $L_i(M)=\lfloor M\log m_1/\log m_i\rfloor$ digits in coordinate $i$ determine an approximate cube. Lemma 3.2 counts these cubes from above and constructs a separated family with the same exponential count. Lemma 3.1 transfers the finite-coordinate count to the dynamical metrics $d_N$ . Dividing by $NM\log m_1$ and using $N^{-1}\log|\pi_i(\Omega)|_N|\to h_{\mathrm{top}}(\pi_i(\Omega),\sigma)$ gives exactly formula (3), including the coefficients $1/\log m_i-1/\log m_{i+1}$ .

Full paper, version 3 ↗

Theorem 1.1, formula (4)Correct

Mean Hausdorff dimension is the weighted entropy divided by $\log m_1$

Pages 3-4 and 10-24 · Theorem 1.1 and Sections 3.2-3.4 · arXiv:2412.02278v3

Lemma 3.3 identifies the recursively defined partition sum $Z_N$ with the weighted topological entropy. The probability measure built from the successive fibre weights yields the exact cylinder-mass identity (11). The Kenyon-Peres scale-selection lemma and the geometric covering lemma give the upper Hausdorff bound, while the uniform concentration estimate, Lemma 3.12's $2^{rN}$ separation bound, and the mass-distribution calculation give the matching lower bound. Both limits equal $\lim_N N^{-1}\log Z_N/\log m_1$ , which is formula (4).

Tsukamoto, two-coordinate prototype and geometric lemmas ↗

Lemma 3.4Correct

Uniformly growing word complexity forces equality of the two dimensions

Pages 12-13 · Lemma 3.4 · arXiv:2412.02278v3

Under the stated uniform fibre-growth hypothesis, the recursive partition function factorizes asymptotically into the projected word complexities with weights $w_a$ . Substitution of the explicit weights converts the weighted-entropy formula into the same entropy combination as formula (3). The uniformity in (9) is exactly what is needed to pass to the $N$ -th roots in every fibre simultaneously.

Full paper, version 3 ↗

02Proofs3 reported findingsCorrect

The approximate-cube count and both Hausdorff bounds are correct and complete. The recursive weighted measure, uniform scale selection, concentration estimate, and separation argument supply all nontrivial steps needed for the two limiting formulas.

Lemmas 3.1-3.2Correct and complete

Finite-coordinate reduction and approximate-cube count

Pages 7-10 · Section 3.1 · arXiv:2412.02278v3

The tail of metric (2) is uniformly negligible after $N_0$ coordinates, giving Lemma 3.1. For Lemma 3.2, the upper cover fixes full words through level $L_r(M)$ and projected words on each remaining block. The lower construction completes each projected word through fixed fibre selections; choosing the lowest coordinate at which two codes differ makes all later completion digits in that coordinate agree, so the corresponding real coordinates differ by at least $m_1^{-M}$ . This verifies both exponential counts used in formula (3).

Full paper, version 3 ↗

Lemmas 3.3 and 3.9; upper bound (15)Correct and complete

Weighted partition sum and the upper Hausdorff estimate

Pages 10-13 and 16-20 · Sections 3.2 and 3.4.1 · arXiv:2412.02278v3

Repeated use of $(x+y)^a\leq x^a+y^a$ for $0\leq a\leq1$ proves the cover characterization of $Z_N$ . The telescoping conditional probabilities defining $\mu_N$ give (11) exactly. Lemma 3.6 applied on the compact family of uniformly Lipschitz sequences supplies one scale $M$ in a bounded interval with the required mass lower bound. The enlarged space $X(N)$ supplies positive-diameter test sets and the cited geometric lemma then yields (15), with the fixed loss $N_0$ disappearing after division by $N$ .

Tsukamoto, source of the covering lemma and prototype argument ↗

Lemmas 3.8, 3.11-3.12; lower bound (19)Correct and complete

Concentration, separation, and the lower Hausdorff estimate

Pages 14-15 and 20-24 · Sections 3.3 and 3.4.2 · arXiv:2412.02278v3

The maximal partial-sum estimate gives a set of $\mu_N$ -measure at least $1/2$ on which every sufficiently fine cylinder has the required upper mass bound. Lemma 3.12 is a finite-grid pigeonhole argument: among more than $2^{rN}$ distinct prefix boxes, two differ by at least two grid units in some coordinate, so their approximate cubes are at distance at least $m_1^{-M}$ . Consequently a set of diameter below that scale meets at most $2^{rN}$ relevant cylinders. Summing their masses over an arbitrary cover gives Hausdorff exponent $\log_{m_1}Z_N-\delta N$ and hence (19).

Full paper, version 3 ↗

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2412.02278v3
Authors listed: Qiang Huo
Audit date: August 18, 2026

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Mean dimension theory for infinite dimensional Bedford-McMullen sponges

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Metric mean dimension is the stated entropy combination

Mean Hausdorff dimension is the weighted entropy divided by log⁡m1\log m_1logm1​

Uniformly growing word complexity forces equality of the two dimensions

Finite-coordinate reduction and approximate-cube count

Weighted partition sum and the upper Hausdorff estimate

Concentration, separation, and the lower Hausdorff estimate

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Mean Hausdorff dimension is the weighted entropy divided by $\log m_1$