On the coincidence of the Hausdorff and box dimensions for some affine-invariant sets

Zhou Feng

On the coincidence of the Hausdorff and box dimensions for some affine-invariant sets

T0 review · 0 major / 2 minor · reviewed 2026-05-24 · grok-4.3

Challenge this review Download PDF Read on arXiv ↗

Pith's one-line read For torus sets invariant under diagonal expansions with at most two distinct rates, Hausdorff and box dimensions coincide exactly when a gauge makes the Hausdorff measure positive and finite and when the maximal entropy measure attains full

desk verdict The paper proves equivalences among Hausdorff-box coincidence, gauge Hausdorff measure, and maximal entropy dimension for affine sets with s≤2 under weak specification, plus counterexamples for s≥3 where the entropy measure still hits Hausdorff dim. read the letter →

arxiv 2405.03213 v1 submitted 2024-05-06 math.DS

Zhou Feng This is my paper

classification math.DS

keywords HausdorffdimensionboxaffineinvariantsetsmaximalentropymeasuregaugefunctionweakspecificationBedford-McMullensponges

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The pith

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

The reading

The paper studies compact invariant sets K in the d-torus under an expanding diagonal endomorphism with s distinct eigenvalues whose symbolic coding obeys weak specification. It establishes that when s is at most 2 the following three properties are equivalent: the Hausdorff dimension of K equals its box dimension, some gauge function makes the Hausdorff measure of K positive and finite, and the measure of maximal entropy supported on K has Hausdorff dimension equal to that of K. When s is at least 3 the paper constructs examples in which the maximal entropy measure is dimensionally full yet the Hausdorff and box dimensions differ. A separate probabilistic argument shows that the first two properties remain equivalent for Bedford-McMullen sponges.

What carries the argument

The three-way equivalence among Hausdorff-box dimension coincidence, existence of a gauge yielding positive finite Hausdorff measure, and full-dimensional maximal entropy measure, for weakly specified symbolic codings when s ≤ 2.

What would settle it

Construct a set K with s=2 and weak specification such that the maximal entropy measure has Hausdorff dimension equal to that of K yet the box dimension strictly exceeds the Hausdorff dimension.

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Extended reading notes

Core claim

When s ≤ 2 the coincidence of Hausdorff and box dimensions of K is equivalent to the existence of a gauge function for which the Hausdorff measure is positive and finite and to the Hausdorff dimension of the measure of maximal entropy equaling the Hausdorff dimension of K, under the weak-specification hypothesis on the symbolic coding.

Load-bearing premise

The symbolic coding of K satisfies weak specification.

Editorial extensions

If this is right

When s ≤ 2, full dimension of the maximal entropy measure forces both dimension coincidence and the existence of a suitable gauge.
When s ≤ 2, dimension coincidence forces both the gauge condition and full dimension of the maximal entropy measure.
For Bedford-McMullen sponges the Hausdorff-box coincidence is equivalent to the gauge condition independently of s.
When s ≥ 3 it is possible for the maximal entropy measure to attain the Hausdorff dimension of K while the box dimension remains strictly larger.

Reading between the lines

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

The change in behavior at s=3 indicates that the number of independent expansion rates can decouple the dimension properties that remain linked in lower-dimensional cases.
The probabilistic method used for sponges may apply to other classes of self-affine sets whose symbolic dynamics lack weak specification.
Counterexamples for s ≥ 3 suggest that any general theory relating these three statements must incorporate the number of distinct eigenvalues as a parameter.

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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.

Desk Editor's Note

read the letter

The main point is that for these compact affine-invariant sets on the d-torus with an expanding diagonal map having s distinct eigenvalues, and assuming weak specification on the symbolic coding, the three statements line up when s≤2: Hausdorff dimension equals box dimension, there is some gauge function making Hausdorff measure positive and finite, and the measure of maximal entropy has dimension equal to that of the set. For s≥3 the authors give examples where the entropy measure still reaches the Hausdorff dimension but the two dimensions of the set itself do not match, a situation they say does not occur in the planar case. They also prove Hausdorff-box equivalence for Bedford-McMullen sponges by a separate probabilistic argument. These equivalences and the higher-s counterexamples are the concrete new pieces. The work sits squarely in the literature on dimensions of self-affine and invariant sets and connects ideas that had been studied separately. The weak specification hypothesis is stated explicitly, so the equivalences are conditional rather than unconditional. The abstract is clear on what is claimed and what is not, and the stress-test note finds no internal gaps in the stated logic. Without the full proofs the details of the constructions cannot be checked here, but nothing in the formulation looks circular or self-referential. This is for people already working on dimension theory for dynamical systems or fractal geometry, especially those dealing with expanding maps or sponges. A reader in that niche would find the equivalences and the s≥3 examples useful to know. The paper deserves a serious referee because the claims are specific, the assumptions are flagged, and the results are presented as new within the subfield.

Referee Report

0 major / 2 minor

Summary. The paper considers compact sets K in the d-torus that are invariant under an expanding diagonal endomorphism with s distinct eigenvalues, assuming the symbolic coding satisfies weak specification. For s ≤ 2 it proves the equivalence of three statements: (A) Hausdorff dimension equals box dimension of K, (B) there exists a gauge function making the Hausdorff measure of K positive and finite, and (C) the Hausdorff dimension of the measure of maximal entropy equals the Hausdorff dimension of K. For s ≥ 3 it constructs examples where (A) fails while (C) holds. Separately, it proves (A) ⇔ (B) for Bedford-McMullen sponges via a probabilistic argument.

Significance. If the stated equivalences and counterexamples hold, the work meaningfully extends the study of dimension coincidence from the planar (s=2) setting to higher-dimensional affine-invariant sets, isolating a new phenomenon for s ≥ 3. The explicit conditioning on weak specification and the probabilistic treatment of sponges constitute concrete technical contributions that can be checked against the hypotheses.

minor comments (2)

[Abstract] The abstract refers to 'some gauge function' without indicating the precise class (e.g., doubling gauges or functions of the form r^α φ(r)); a brief clarification in the introduction would aid readability.
[Introduction] The statement of the counterexamples for s ≥ 3 would benefit from an explicit reference to the section containing the construction, even if only a high-level outline appears in the introduction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its contributions, and recommendation of minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The paper proves equivalences (A)⇔(B)⇔(C) for s≤2 explicitly conditioned on the external assumption that the symbolic coding satisfies weak specification; it separately constructs counterexamples for s≥3 and applies an independent probabilistic argument to Bedford-McMullen sponges. No derivation step reduces by the paper's own equations to a fitted parameter, self-referential definition, or load-bearing self-citation chain; all central claims rest on stated external hypotheses and distinct methods rather than internal re-labeling of inputs.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The central claims rest on the domain assumption of weak specification for the symbolic dynamics and on the geometric setup of the expanding diagonal endomorphism; no free parameters or invented entities are introduced.

assumptions (2)

domain assumption The symbolic coding of K satisfies weak specification.
Invoked as the key supposition enabling the equivalence theorem for s ≤ 2.
domain assumption K is compact, torus-invariant under an expanding diagonal endomorphism with s distinct eigenvalues.
Defines the class of sets under study.

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Pith. "Pith review of On the coincidence of the Hausdorff and box dimensions for some affine-invariant sets." pith.science (2026). https://pith.science/paper/2405.03213

@misc{pith2026240503213,
  author       = {Pith},
  title        = {Pith review of: On the coincidence of the Hausdorff and box dimensions for some affine-invariant sets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2405.03213}},
  note         = {Machine review of arXiv:2405.03213}
}

abstract

Let $ K $ be a compact subset of the $d$-torus invariant under an expanding diagonal endomorphism with $ s $ distinct eigenvalues. Suppose the symbolic coding of $K$ satisfies weak specification. When $ s \leq 2 $, we prove that the following three statements are equivalent: (A) the Hausdorff and box dimensions of $ K $ coincide; (B) with respect to some gauge function, the Hausdorff measure of $ K $ is positive and finite; (C) the Hausdorff dimension of the measure of maximal entropy on $ K $ attains the Hausdorff dimension of $ K $. When $ s \geq 3 $, we find some examples in which (A) does not hold but (C) holds, which is a new phenomenon not appearing in the planar cases. Through a different probabilistic approach, we establish the equivalence of (A) and (B) for Bedford-McMullen sponges.

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Pith tools

Reviewed May 24, 2026 · model on record in the stance chip above.

[1] [1]

Hausdorﬀ dimension of the limit sets of som e planar geometric constructions

Krzysztof Bara´ nski. Hausdorﬀ dimension of the limit sets of som e planar geometric constructions. Adv. Math. , 210(1):215–245, 2007

work page 2007

[2] [2]

Non-uniqueness of ergodic meas ures with full Haus- dorﬀ dimensions on a Gatzouras-Lalley carpet

Julien Barral and De-Jun Feng. Non-uniqueness of ergodic meas ures with full Haus- dorﬀ dimensions on a Gatzouras-Lalley carpet. Nonlinearity, 24(9):2563–2567, 2011

work page 2011

[3] [3]

Barreira

Luis M. Barreira. A non-additive thermodynamic formalism and app lications to dimension theory of hyperbolic dynamical systems. Ergodic Theory Dynam. Systems, 16(5):871–927, 1996

work page 1996

[4] [4]

Crinkly curves, markov partitions and dimension

Tim Bedford. Crinkly curves, markov partitions and dimension. University of War- wick, 1984

work page 1984

[5] [5]

A course in probability theory

Kai Lai Chung. A course in probability theory . Academic Press, Inc., San Diego, CA, third edition, 2001

work page 2001

[6] [6]

The Hausdorﬀ and dynamical dimen sions of self- aﬃne sponges: a dimension gap result

Tushar Das and David Simmons. The Hausdorﬀ and dynamical dimen sions of self- aﬃne sponges: a dimension gap result. Invent. Math. , 210(1):85–134, 2017

work page 2017

[7] [7]

Probability—theory and examples , volume 49 of Cambridge Series in Statistical and Probabilistic Mathematics

Rick Durrett. Probability—theory and examples , volume 49 of Cambridge Series in Statistical and Probabilistic Mathematics . Cambridge University Press, Cambridge, ﬁfth edition, 2019

work page 2019

[8] [8]

Falconer

Kenneth J. Falconer. The Hausdorﬀ dimension of self-aﬃne frac tals. Math. Proc. Cambridge Philos. Soc. , 103(2):339–350, 1988

work page 1988