arXiv:2606.03941v2

Rate distortion dimension of Gibbs measures for functions depending on the first two coordinates on the full shift of Ahlfors regular spaces

Kanji Inui, Mao Shinoda

math.DS37A0537D3594A3437C30

Paper metadata

Original submission: June 2, 2026
Latest revision: August 10, 2026 · version 2
Claimed MSC: 37A05, 37D35, 94A34, 37C30
arXiv categories: math.DS
Paper license: arXiv non-exclusive distribution license ↗

Abstract

The study on shift spaces in ergodic theory has extended beyond the classical setting, but there is room to discuss an extension of the Kolmogorov-Sinai entropy in the ergodic theoretical point of view. On the other hand, the rate distortion dimension recently attracted attention in mean dimension theory because it behaves like the Kolmogorov-Sinai entropy on dynamical systems in the ``large" spaces in which the usual entropy is in general infinite. According to these background, we investigate the connection between the Gibbs measure on the product spaces and the variational principle based on the rate distortion dimension. Concretely we calculate the rate distortion dimension of the Gibbs measure for a Hölder continuous potential depending on the first two coordinates and it satisfies the simplest case of themodynamic formalism based on the rate distortion dimension: the extension of the maximal measure of topological entropy. Notably, to the best of our knowledge, this is the first study to introduce Gibbs measures into mean dimension theory, and this connection allows the mean dimension with potential to be computed more naturally through the notion of the zero-temperature limit, in the spirit of thermodynamic formalism.

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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 18, 2026

01Statements3 reported findingsContains unsupported statements

The mean-dimension-with-potential formula in Theorem 2 is correct. The rate-distortion formula for Gibbs measures in Theorem 1, and the variational corollaries that depend on it, are not able to be verified because the lower-bound proof uses a false Markov assertion about a quantized process.

Theorem 1Not able to verify

The rate-distortion dimension formula is not verified

Pages 3 and 15-19 · Theorem 1 and Section 3.2 · arXiv:2606.03941v2

The upper bound $\operatorname{rdim}(\mu_\phi)\leq s$ follows from the general variational inequality and Proposition 2.4. The reverse bound relies on replacing the joint entropy of the quantized state sequence by $H(\widetilde X_0)+(N-1)H(\widetilde X_1\mid\widetilde X_0)$ . An arbitrary measurable quantization of a Markov chain is generally a hidden Markov process, not a first-order Markov chain, so that replacement is unsupported and has the wrong inequality direction for the desired lower bound. No counterexample to the formula itself was established, but a new entropy-rate argument is required.

Full paper, version 2 ↗

Corollaries 1.2 and 1.4Not able to verify

The variational conclusions inherit the unverified lower bound

Pages 4-5 · Corollaries 1.2 and 1.4 · arXiv:2606.03941v2

Both corollaries use Theorem 1 to assert that every displayed Gibbs measure has rate-distortion dimension $s$ . Their remaining inputs give the ambient mean metric dimension and the maximizing zero-temperature limit, but do not independently supply that measure-theoretic equality. Therefore the attainment and potential-weighted conclusions remain unverified to exactly the same extent as Theorem 1.

Full paper, version 2 ↗

Theorem 2Correct

Mean dimension with potential of the full cube shift

Pages 5-6 and 14-15 · Theorem 2 and Section 3.1 · arXiv:2606.03941v2

Projection to the first $N+k+1$ cube coordinates is a $2^{-k}$ -embedding and gives the upper bound $D+\beta(\phi)$ . For the lower bound, choose an orbit segment whose Birkhoff average approaches $\beta(\phi)$ , vary its first $n$ cube coordinates in small full-dimensional boxes, and use uniform continuity of $\phi$ together with the width-dimension lemma. Division by $n$ and passage to the limits gives $\operatorname{mdim}((([0,1]^D)^{\mathbb N_0},\sigma,\phi))=D+\sup_\mu\int\phi\,d\mu$ .

Full paper, version 2 ↗

02Proofs5 reported findingsContains incorrect or incomplete proofs

The proof of Theorem 1 contains a decisive entropy gap: coordinatewise quantization does not preserve the Markov property. Two local constant computations in that proof and one one-sided-index expression in the proof of Theorem 2 have mechanical corrections and do not affect the substantive verdicts.

Proof of Theorem 1Incomplete as written

The quantized process need not be Markov

Page 17 · first entropy estimate after the data-processing inequalities · arXiv:2606.03941v2

Let $\widetilde X_i=p(X_i)$ for the finite partition $p$ . The proof invokes the Markov property of $(X_i)$ to use $H(\widetilde X_0,\ldots,\widetilde X_{N-1})=H(\widetilde X_0)+(N-1)H(\widetilde X_1\mid\widetilde X_0).$ This identity is false without a lumpability condition: a function of a Markov chain is generally hidden Markov, and its next-symbol law can depend on the whole observed past. Indeed, chain rule only gives conditional entropies with the full past, which are at most the one-step conditional entropy; substituting the latter therefore increases the term that the proof needs as a lower bound. Repair classification: No repair supplied. One must bound the entropy rate conditioned on the full quantized past or use a different rate-distortion lower-bound argument.

Full paper, version 2 ↗

Equation (23)Typo

The lower density constant is missing a factor

Page 18 · Equation (23) · arXiv:2606.03941v2

With $\theta=\psi\overline\psi/\pi$ and $\psi_{\min}=\min\{\min\psi,\min\overline\psi\}$ , the displayed lower coefficient should be $\psi_{\min}^2/\pi$ , not $\psi_{\min}$ . Replacing the coefficient by this positive constant is mechanically forced by the definition of $\theta$ and only changes the additive constant denoted by $C$ ; it has no effect on the limiting rate-distortion exponent.

Full paper, version 2 ↗

Equation (25)Typo

The sign of the Ahlfors constant is reversed in the simplification

Page 18 · last line of Equation (25) · arXiv:2606.03941v2

From the preceding line, the $n$ -dependent term simplifies to $C+(n-1)s\log\kappa-(n+1)\log c+\log\rho_L$ , not $C+(n-1)\log(c\kappa^s)+\log\rho_L$ . The correction follows by expanding $-\log(c\kappa^s)-n\log(c\kappa^{-s})$ . After division by $(n+1)\log\kappa$ , the corrected bound still tends to $s-\log c/\log\kappa$ , and then to $s$ as $\kappa\to\infty$ ; thus this local sign error does not create the substantive gap in Theorem 1.

Full paper, version 2 ↗

Proof of Theorem 2Typo

The negative-coordinate factor should be omitted

Page 15 · definition of $Y_n$ · arXiv:2606.03941v2

The system is the one-sided shift indexed by $\mathbb N_0$ , but the displayed definition of $Y_n$ begins with $\prod_{i=-\infty}^{-1}\{z_i\}$ . Those coordinates do not exist. Omitting that factor gives $Y_n=\prod_{i=0}^{n-1}I_i\times\prod_{i=n}^{\infty}\{z_i\}$ , which is exactly the set used in the subsequent metric and width-dimension calculation. The repair is unique and harmless.

Full paper, version 2 ↗

Proof of Theorem 2Correct and complete

Embedding and orbit-box bounds

Pages 14-15 · Section 3.1 · arXiv:2606.03941v2

After the one-sided-index typo is corrected, the finite-coordinate projection is an appropriate embedding, Proposition 3.1 supplies the maximal ergodic average, and uniform continuity controls the potential on each orbit box. The cited cube width-dimension identity applies to the product of nondegenerate coordinate intervals, so the upper and lower bounds meet at $D+\beta(\phi)$ .

Full paper, version 2 ↗

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2606.03941v2
Authors listed: Kanji Inui, Mao Shinoda
Audit date: August 18, 2026

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Rate distortion dimension of Gibbs measures for functions depending on the first two coordinates on the full shift of Ahlfors regular spaces

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The rate-distortion dimension formula is not verified

The variational conclusions inherit the unverified lower bound

Mean dimension with potential of the full cube shift

The quantized process need not be Markov

The lower density constant is missing a factor

The sign of the Ahlfors constant is reversed in the simplification

The negative-coordinate factor should be omitted

Embedding and orbit-box bounds

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