Mean Assouad dimension and spectrum, with applications to infinite dimensional fractals

Adam \'Spiewak; Qiang Huo

REVIEW 4 major objections 4 minor 7 references

Mean Assouad dimension and spectrum, with applications to infinite dimensional fractals

T0 review · 4 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read The paper introduces the mean Assouad dimension and spectrum as new bi-Lipschitz invariants of dynamical systems, and derives closed-form formulae for the mean Assouad dimension and spectrum of every Bedford-McMullen carpet system in terms

desk verdict A genuinely new mean Assouad invariant with a clean carpet formula, but the main theorem rests on an omitted proof (Lemma 5.7) that a referee should demand before the result is treated as established. read the letter →

arxiv 2601.00233 v2 pith:VVXK6U3I submitted 2026-01-01 math.DS math.CA

Qiang Huo , Adam Śpiewak This is my paper · ORCID

classification math.DSmath.CA MSC 28A8037B4037C4537B10

keywords meanAssouaddimensionspectruminterpolationBedford-McMullencarpetsystemstopologicalconditionalentropybi-Lipschitzinvariancemetricinfinite-dimensionalfractals

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

The paper defines a dynamical analogue of the Assouad dimension—the mean Assouad dimension—and its one-parameter interpolation, the mean Assouad spectrum, for a compact dynamical system with a metric. These are new bi-Lipschitz invariants that sit between the metric mean dimension and the classical Assouad dimension. The central result is that, for every Bedford-McMullen carpet system, both quantities have closed-form formulae: the mean Assouad dimension is the topological conditional entropy of the factor map divided by log a plus the topological entropy of the projected base divided by log b. A surprising consequence is that the mean Assouad spectrum undergoes a phase transition at the scale ratio log b/log a. This gives a concrete way to measure the 'thickest' scaling behaviour of infinite-dimensional fractals, completing a program of computing mean-type dimensions for these systems.

What carries the argument

The central object is S(X,r,ρ)=lim_{M→∞}(1/M)sup_{x∈X}log N_{d_M}(B_{d_M}(x,r),ρ), the growth rate in orbit length of how many ρ-balls cover an M-step Bowen ball of radius r. Sub-additivity in M (Proposition 2.1) guarantees the limit, and the paper defines mdim_A as the infimum s with e^{S(X,r,ρ)}≤C(r/ρ)^s uniformly in scales; the spectrum fixes ρ=r^{1/θ}. For carpets, the proof switches to finite coordinate block covers by 'approximate squares' and counts words in Ω|_N above each base word in Ω'|_N – this is where the conditional entropy h_top(Ω|Ω',σ) enters. The phase transition at θ=log b/log a is driven by the two scale levels l_1(r), l_2(r) defined by a^{-l_1}≤r<a^{-l_1+1}, b^{-l_2}≤r<b

What would settle it

Compute, for a carpet system with non-uniform fibres (so h_top(Ω|Ω',σ)>0), the quantity S(X,r,ρ) directly from the definition using the weighted metric d of (5.1), or via a discrete simulation of the N-truncated systems; then compare the resulting slope with h_top(Ω|Ω',σ)/log a + h_top(Ω',σ)/log b. A disagreement, or even a measured dependence of the limit on the choice of the weight sequence in (5.1), would falsify the theorem as stated.

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Extended reading notes

Core claim

For a Bedford-McMullen carpet system X_Ω with integer contraction ratios a > b ≥ 2, the paper proves that mdim_A(X_Ω,σ,d) = h_top(Ω|Ω',σ)/log a + h_top(Ω',σ)/log b, where Ω is the defining subshift, Ω' its projection onto B^N, h_top(Ω|Ω',σ) is the topological conditional entropy of the factor map, and h_top(Ω',σ) is the entropy of the base. For the spectrum, the paper gives an interpolating formula for θ∈(0, log b/log a] that starts from the metric mean dimension at θ=0 and increases to the mean Assouad dimension at θ=log b/log a, and shows that for θ∈(log b/log a,1) the spectrum is constant, equal to the mean Assouad dimension. Thus the entire scale-dependence of this extreme dimension is c

Load-bearing premise

The proof of Theorem 5.2 is carried out entirely in the ℓ∞ picture on finite coordinate projections, but the theorem is stated for the weighted product metric d of (5.1); the paper states Lemma 5.7 – that the two give the same mean Assouad dimension – without proof, and the formula collapses if this equivalence fails.

Editorial extensions

If this is right

For every Bedford-McMullen carpet system, the mean Assouad dimension and full spectrum are now computed in closed form, reducing the problem to two topological entropies and the ratio log b/log a.
The mean Assouad spectrum is a genuine interpolation: it starts at the metric mean dimension when θ→0, rises monotonically, and becomes flat at the mean Assouad dimension for all θ≥log b/log a.
A carpet system has equal metric mean dimension and mean Assouad dimension exactly when h_top(Ω,σ)=h_top(Ω',σ)+h_top(Ω|Ω',σ), the dynamical analogue of 'uniform fibres'.
The ratio log b/log a is a bi-Lipschitz invariant of these carpet systems, so systems with different scale ratios cannot be bi-Lipschitz conjugate in this class.
For full shifts on a compact alphabet, the mean Assouad dimension and spectrum equal the corresponding Assouad dimension and Assouad spectrum of the alphabet.

Reading between the lines

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

If the omitted Lemma 5.7 is filled in, the same approximate-square counting should work for higher-dimensional sponge systems, with the conditional entropy terms replaced by the appropriate fibre entropies; the phase-transition structure would survive.
The flatness of the spectrum above log b/log a suggests that, for very large outer scales relative to the anisotropy, the most 'Assouad-like' behaviour is governed purely by the number of vertical fibres and the base entropy, and is insensitive to the fibre complexity – a phenomenon that could be tested numerically.
Since classical Assouad dimension controls almost bi-Lipschitz embeddings, the new mean invariant may provide sharper obstructions in mean-dimension embedding theory when the classical Assouad dimension is large.
A testable extension: approximate the carpet system by its finite N-coordinate truncations and measure the empirical S(X,r,ρ); the predicted formula should hold uniformly in r and ρ once the number of coordinates grows as the approximate-square argument requires.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

What you should know: Huo and Śpiewak introduce a mean Assouad dimension and spectrum for dynamical systems, and they compute both in closed form for Bedford–McMullen carpet systems. The main theorem, Theorem 5.2, gives mdim_A = h_top(Ω|Ω′,σ)/log a + h_top(Ω′,σ)/log b and an interpolating spectrum with the expected phase transition at log b/log a. This is a real contribution: these invariants are new, the carpet formulas go beyond Tsukamoto's mean Hausdorff and metric mean dimension results, and the formulas reduce correctly to the classical planar Bedford–McMullen dimension in the finite-coordinate picture.

The paper does several things well. The bi-Lipschitz invariance is straightforward and correctly stated. The full-shift computation, Proposition 4.1, reduces the mean Assouad dimension to the Assouad dimension of the alphabet without a variational principle, and the band-limited function example is a nice sanity check. The comparison in Section 6 with alternative definitions is honest and useful. The covering arguments in §5.3–5.5 are detailed and, as far as I can tell, the counting is consistent with the known non-dynamical estimates.

The soft spots are real but not all equal. The biggest is Lemma 5.7: the equivalence between the original metric (5.1) and the ℓ∞-projection quantity is stated without proof, with the claim that it is essentially Lemma 4.2. Lemma 4.2 is proved, but it is for a full shift on a product alphabet; the carpet system has a non-injective digit coding and a weighted metric, so the same proof needs to be checked, not waved at. Since Theorem 5.2 is computed entirely in the ℓ∞ picture, this is load-bearing. I would not call it fatal—the comparability estimates look plausible—but the current manuscript does not establish the theorem as written.

The stress-test concern about non-unique digit codings does not, on reflection, land. Each coordinate has at most two base-a and two base-b expansions, so the coding map has at most 4^N preimages per point. That is an additive constant in the exponential growth rate, which the dimension definition absorbs. It may complicate exact covering-number identities, but it should not change the formula.

Two smaller issues: the paper defines systems as homeomorphisms but then studies one-sided shifts, which are not invertible; this is easy to fix but as written it is a mismatch. And the proof of Proposition 3.5 has at least one inclusion after the partition (3.9)–(3.10) that I could not justify from the text. That result is peripheral to the main theorem, but it deserves a careful check.

Bottom line: this is a serious paper with a new invariant and a nontrivial computation. It deserves peer review, not rejection. The referee should ask for a full proof of Lemma 5.7, a cleanup of Proposition 3.5, and a word on whether the theory is meant for continuous maps rather than homeomorphisms.

Referee Report

4 major / 4 minor

Summary. The paper introduces a mean Assouad dimension and a mean Assouad spectrum for topological dynamical systems, defined by a dynamical analogue of the Assouad dimension using Bowen balls. It proves bi-Lipschitz invariance, bounds relating the spectrum to metric mean dimension, and a reduction to the non-wandering set. It computes the new invariants for full shifts (in terms of the Assouad dimension of the alphabet) and for the space of band-limited functions (where the metric mean dimension and mean Assouad dimension coincide). The main application is Theorem 5.2, which gives explicit closed-form formulas for the mean Assouad dimension and spectrum of infinite-dimensional Bedford–McMullen carpet systems in terms of the topological conditional entropy and the entropy of the base subshift. The proof of Theorem 5.2 is carried out via a reduction to an ℓ∞ metric on finite projections and a counting of approximate squares.

Significance. If the main theorem and the supporting general results are correct, the paper opens a new direction in the mean-dimension program by importing Assouad-type scaling ideas into dynamical systems. The carpet-system formulas are explicit and show a phase transition at θ = log b/log a, in analogy with the planar theory. The new invariants are natural and the paper provides several test calculations. However, the current version does not fully prove the central theorem because a key reduction lemma is stated without proof, and the lower-bound estimates are not rigorously justified. Several general statements also contain proof gaps.

major comments (4)

[§5.2, Lemma 5.7] Lemma 5.7, which equates the mean Assouad dimension of the carpet system under the weighted metric (5.1) with the ℓ∞-projection quantity, is not proved; the text says the proof is 'essentially the same' as Lemma 4.2. This is load-bearing because all subsequent estimates in §5.3–5.5 are performed in the ℓ∞-picture. The analogy with Lemma 4.2 is not automatic: in the full-shift case the coding map is one-to-one on coordinates, while for carpet systems the map from (A×B)^N to X_Ω is many-to-one because of non-unique base-a and base-b expansions. Distinct coding choices can represent the same geometric point, so the covering number of a geometric set can be smaller than the number of coding choices. A full proof of Lemma 5.7 is needed, and it must address this multiplicity (e.g., by showing the coding is injective on a set of full dimension relevance or that the multiplicity is uniformly bou
[§5.4–5.5, lower bound] The lower bounds for the mean Assouad dimension and spectrum are not rigorously established. After Eq. (5.7), the paper states that 'the unique inequality in (5.7) is replaced by equality' and that the covering estimates are 'optimal up to multiplicative constants.' But (5.7) is an upper bound; to obtain a lower bound one must exhibit a separated collection of ρ-balls or prove that the counted approximate squares are pairwise distinct geometric sets. The current argument counts coding choices, and if many-to-one collapses occur, the true covering number may be smaller. The authors should provide a direct lower-bound argument in the geometric space, or restrict to points with unique expansions and verify that this does not change the mean Assouad dimension.
[§3.2, Eq. (3.2)–(3.3)] The proof of the inequality mdim_M(X,T,d) ≤ mdim^θ_A(X,T,d) is flawed. The definition of mdim^θ_A provides bounds only for scale pairs of the form (r, r^{1/θ}). The proof applies this to the pair (ε^{i/θ}, ε^{(i+1)/θ}), which is of that form only when θ = i/(i+1), not for all i. Consequently the bound on sup_x N(B(x, ε^{i/θ}), ε^{(i+1)/θ}) does not follow from the definition. This step is essential for Proposition 3.2 and Corollary 3.3. A correct chaining argument is needed, or the statement must be modified.
[§3.4, line after Eq. (3.10)] The inclusion T^{min J}(B_{d_M}(x,r) ∩ X_I) ⊂ B_{d_{|J|}}(T^{min J}x, r) ∩ Ω_{|J|,ρ} is not justified as written. For y in this set and i ∈ J, we have dist(Ω, T^i y) ≤ dist(Ω, T^i x) + d(T^i x, T^i y) < ρ + r, not < ρ. The subsequent application of Lemma 3.7 (which gives a bound using Ω_{|J|,ρ}) is therefore invalid. The constants need to be readjusted (e.g., using a neighborhood of radius ρ+r and a correspondingly larger covering scale) or a different proof is required.

minor comments (4)

[§4, heading] Typo: 'dimesnion' should be 'dimension'.
[§5.4, §5.5] Typo: 'it we choose' should be 'if we choose' in two places.
[§5.2, §5.5] Notation for the projected alphabet is inconsistent: |Ω'_N| in §5.5 should be |Ω'|_N| to match the notation introduced in §5.2.
[§3.2, proof of Proposition 3.2] The constant C_2 in the proof appears as C_2^M but is not explicitly related to the constant in the definition of mdim^θ_A; this is a minor clarity issue.

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity in the main derivation; formula follows from covering estimates with intrinsic entropies. A minor self-citation appears but is not load-bearing.

full rationale

The central derivation is self-contained. The main formula (5.3) is not assumed: Theorem 5.2 is obtained by explicit covering-number estimates on approximate squares, and the entropy terms h_top(Ω|Ω',σ) and h_top(Ω',σ) enter as intrinsic symbolic rates (via Lemma 5.6) rather than as fitted parameters. The upper bound (Claim 5.8) is a direct counting argument, the lower bound in §5.4 uses the same estimates with optimal fibers, and the spectrum formula in §5.5 follows from the same covering count together with the scale comparison in Lemma 5.9. Lemma 5.7 is indeed stated without proof ('We omit the proof of Lemma 5.7 since it is essentially the same to that of Lemma 4.2'), but this is an omitted proof / completeness gap, not a circularity: the claimed equality between the metric (5.1) version and the ℓ∞-projection version is not built into the definition of mdim_A(X_Ω,σ,d), and Lemma 4.2, which it is said to mimic, is proved in full. The self-citation [GŚ20] appears only as inspiration for Lemma 4.2 and as an auxiliary example in Section 6; it does not carry Theorem 5.2. No fitted input is renamed as a prediction, and no uniqueness theorem is imported from the authors' prior work. Because the self-citation is minor and not load-bearing, the circularity score is 2 rather than 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

The central claim rests on standard subadditivity arguments, two external stable-set lemmas, the unproved Lemma 5.7 bridging the weighted metric to the ℓ∞ picture, a comparability result from Fraser's book for approximate squares, and an implicit extension of the theory from homeomorphisms to continuous non-invertible maps. All are domain assumptions typical for this area except Lemma 5.7, which is effectively an omitted proof.

assumptions (5)

standard math Fekete's lemma for subadditive sequences
Used to guarantee limits S(X,r,ρ) exist in Prop 2.1 and topological entropy limits; standard.
domain assumption [Tsu22, Prop 2.2] and [Bow72, Prop 2.2]: stable set covers lift to Bowen ball covers
Used in Prop 3.4 to characterize mdim_A via stable sets; not proved in the paper.
ad hoc to paper Lemma 5.7: mean Assouad dimension of the carpet system with weighted metric equals the ℓ∞-metric finite-projection version
Explicitly stated without proof; identical in spirit to Lemma 4.2 whose proof is given. Load-bearing for Theorem 5.2.
domain assumption Comparability of Bowen balls and approximate squares in the carpet (cf. [Fra14, Lemma 7.1])
Stated in §5.2 to justify replacing balls by Q_{N,r}; cited to Fraser's book/paper, not proved.
domain assumption The framework extends to non-invertible continuous maps (one-sided shifts)
Definitions in §2.2 assume homeomorphisms, but main examples are one-sided shifts; the paper never reconciles this.

invented entities (2)

Mean Assouad dimension mdim_A
purpose: New dynamical invariant measuring thickness of orbit segments at two scales
Introduced in §2.3; its value is demonstrated by internal examples, but there is no external benchmark or falsifiable prediction outside this paper.
Mean Assouad spectrum mdim^θ_A
purpose: One-parameter family of invariants interpolating between metric mean dimension and mean Assouad dimension
Introduced in §2.3; like mdim_A, evidence is internal to the paper.

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Cite this review

Pith. "Pith review of Mean Assouad dimension and spectrum, with applications to infinite dimensional fractals." pith.science (2026). https://pith.science/paper/VVXK6U3I

@misc{pith2026260100233,
  author       = {Pith},
  title        = {Pith review of: Mean Assouad dimension and spectrum, with applications to infinite dimensional fractals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VVXK6U3I}},
  note         = {Machine review of arXiv:2601.00233}
}

read the original abstract

We introduce the mean Assouad dimension of a dynamical system, motivated by the Assouad dimension in fractal geometry. Using dimension interpolation, we further define the mean Assouad spectrum. This provides a new family of bi-Lipschitz invariants of dynamical systems. We study its basic properties and calculate it for several classes of dynamical systems. As an application, we determine explicit formulae for the mean Assouad dimension and spectrum of infinite-dimensional Bedford--McMullen carpet systems, contributing to the program of studying infinite dimensional fractals, initiated recently by Tsukamoto.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

7 extracted references · 5 linked inside Pith

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

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