Pith. sign in

REVIEW 2 major objections 4 minor 22 references

Assouad and lower dimensions of graph-directed Bedford-McMullen carpets

T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves exact formulas for the Assouad and lower dimensions of every graph-directed Bedford-McMullen carpet, expressed as growth rates of explicit counting sequences, and identifies exactly when box, Hausdorff, and Assouad…

desk verdict The main dimension formulas are new and well argued; the box-Assouad coincidence theorem leans on an unreviewed preprint, and the counterexample is unverified—both fixable. read the letter →

arxiv 2411.15407 v1 pith:3SVDJSQU submitted 2024-11-23 math.CA

Hua Qiu , Qi Wang , Shufang Wang This is my paper · ORCID
classification math.CA MSC 28A8028A78
keywords Assouaddimensionlowerdirectedgraphself-affinesetBedford-McMullencarpetgraph-directediteratedfunctionsystemboxHausdorff
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

Graph-directed Bedford-McMullen carpets are self-affine fractals built by placing $n$-by-$m$ rectangular blocks according to the edges of a finite directed graph, contracting horizontally by $1/n$ and vertically by $1/m$. The paper proves that two numbers, $\alpha$ and $\beta$, computed from the box dimensions of the vertical projections of the pieces of the construction, give the exact Assouad and lower dimensions: $\dim_A X = \log \alpha / \log n$ and $\dim_L X = \log \beta / \log n$. This holds without the rectangular open set condition, so overlapping contractions are allowed. The paper also identifies an equivalent condition, (1.3), for the box and Assouad dimensions to coincide, and shows that under that condition the Hausdorff dimension takes the same common value. A constructed example shows that the previously observed all-equal or all-distinct dichotomy fails for non-irreducible graphs, where $\dim_L X < \dim_H X = \dim_B X < \dim_A X$ is possible.

What carries the argument

The machinery is the pair of growth rates $\alpha$ and $\beta$ together with approximate-square counting. An approximate square of level $k$ is a rectangle of width $n^{-\lfloor k\log_n m\rfloor}$ and height $m^{-k}$, the natural nearly square cell at that scale; the ratio $n>m$ is why one index is floored. The sequences $\alpha_k$, $\beta_k$ are defined from weighted counts of equivalence classes of admissible words: $\eta(v,x_w,y_w)$ records the box dimension of the vertical projection of the piece of $X_v$ lying in $\psi_w((0,1)^2)$, so the weight $n^{k\eta}$ converts a projection dimension into a horizontal counting factor. The upper and lower bounds for $\dim_A X$ reduce to showing that the number $N_{k'}(X\cap Q_k^\circ(p,q))$ of level-$k'$ approximate squares needed to cover the part of $X$ in a level-$k$ square grows like $m^{(k'-k)(\log\alpha/\log n+\epsilon)}$, and Lemmas 3.2 and 3.3 establish both directions; Lemmas 4.1 and 4.2 do the same for $\log\beta/\log n$. For the dimension-comparison theorems, the same entropies are expressed through $h_{\mathrm{top}}(I)$, $h_{\mathrm{top}}(\pi I)$, and the fiber entropies $h_{\mathrm{top}}(\pi^{-1}(y))$, and the coincidence criterion (1.3) is exactly the equality case of the entropy inequality (5.1).

What would settle it

Construct or find an irreducible graph-directed carpet in which two vertical words $y,y'$ of the same length $k$ have fiber counts $\#\{x_w : w\in E_k,\, y_w=y\}$ and $\#\{x_w : y_w=y'\}$ whose ratio grows without bound as $k\to\infty$; then the uniform-fiber premise behind Lemma 5.3 is false, and the coincidence criterion in Theorem 1.3 would have to be re-tested.

Watch

Extended reading notes

Core claim

On its own terms, the paper's discovery is that the extreme local scaling rates of any graph-directed Bedford-McMullen carpet family are completely governed by the submultiplicative sequences $\{\alpha_k\}$ and $\{\beta_k\}$ of Section 2.2. For each level $k$, $\alpha_k$ is the maximum, over start vertices $v$ and vertical words $y$ of length $k$, of a sum over equivalence classes $[w]$ of admissible words with vertical projection $y$, each class weighted by $n^{k\eta(v,x_w,y_w)}$, where $\eta$ is the box dimension of the vertical projection of the piece $X_v \cap \psi_w((0,1)^2)$. The number $\alpha = \lim_k (\alpha_k)^{1/k}$, and $\beta = \liminf_k (\beta_k)^{1/k}$ is defined through the analogous minimum-based sequence $\beta_k$. Theorem 1.1 asserts $\dim_A X = \log \alpha / \log n$ and $\dim_L X = \log \beta / \log n$. The proof replaces weak-tangent constructions by counting approximate squares, and the later theorems translate the same counts into topological-entropy form: in the irreducible case $\dim_A X = h_{\mathrm{top}}(\pi I)/\log m + \sup_{y\in\pi I} h_{\mathrm{top}}(\pi^{-1}(y))/\log n$, with box dimension agreeing with Assouad dimension exactly when (1.3) holds, in which case $\dim_H X = \dim_B X = \dim_A X$ as well.

Load-bearing premise

The load-bearing premise is the uniform fiber-counting estimate cited as [8]: in an irreducible graph-directed system, every vertical projection word of length $k$ is realized by roughly the same number of admissible words, with the comparison constant independent of the word and of $k$; if that estimate fails, the equivalence between $\dim_H X=\dim_B X$ and $\dim_B X=\dim_A X$ loses its support.

Editorial extensions

If this is right

  • For every graph-directed Bedford-McMullen carpet, the Assouad and lower dimensions are now determined by the explicit sequences $\alpha_k$ and $\beta_k$, even when different contractions have identical images.
  • The formulas bypass weak-tangent constructions, so the same counting scheme should apply to carpet families whose tangent structure is difficult to build.
  • Condition (1.3) gives a checkable criterion: the box and Assouad dimensions coincide exactly when one irreducible component realizes the maximum in the box-dimension formula and its fiber entropies sum to $h_{\mathrm{top}}(I)$.
  • Whenever the box and Assouad dimensions coincide, the Hausdorff dimension is forced to the same value, so the three global dimensions collapse.
  • In the non-irreducible case the clean dichotomy is false: the paper's $4\times3$ example satisfies $\dim_L X=1$, $\dim_H X=\dim_B X=3/2$, and $\dim_A X=2$.

Reading between the lines

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

  • An immediate testable extension is to implement the $\alpha_k$ and $\beta_k$ sequences numerically for a given graph: truncations give rigorous upper and lower bounds on $\alpha$ and $\beta$, hence on $\dim_A$ and $\dim_L$.
  • The same approximate-square counting likely extends to the more flexible grid class of box-like carpets and to higher-dimensional self-affine sponges, where the graph-directed version of these formulas has not yet been carried out.
  • The non-irreducible counterexample suggests that dimension coincidence is controlled by how irreducible components feed into one another; a decomposition into a component together with its reachable future may be the right object for a full classification.
  • The coincidence criterion depends on the uniform fiber-counting theorem cited as [8]; if that theorem changes, only the equivalence part of Theorems 1.3 and 1.4 would need revision, not the $\alpha,\beta$ formulas themselves.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper studies the Assouad and lower dimensions of graph-directed Bedford–McMullen (×m,×n)-carpets, i.e. self-affine sets generated by a finite directed graph with contractions of the form (ξ1,ξ2) ↦ (n^{-1}(ξ1+x_e), m^{-1}(ξ2+y_e)). In Section 2.2 it introduces sequences α_k and β_k built from the box dimensions of projections of construction pieces, and Theorem 1.1 asserts dim_A X = log α/log n and dim_L X = log β/log n, without assuming the rectangular open set condition. Section 3 proves the Assouad formula via approximate-square estimates; Section 4 proves the lower-dimension formula. Section 5 compares box and Assouad dimensions: Theorem 1.3 gives a general formula for dim_A in terms of irreducible components and entropies, an equivalent condition for dim_B X = dim_A X, and the conclusion that in that case dim_H X = dim_B X = dim_A X. Theorem 1.4 claims that for irreducible G, dim_H=dim_B implies dim_B=dim_A, and supplies an example showing the analogue fails when G is not irreducible.

Significance. If correct, these results settle the Assouad and lower dimensions for a broad class of graph-directed planar carpets and give a sharp coincidence criterion relating box, Hausdorff, and Assouad dimensions, extending the dichotomy found by Mackay and Fraser. The main estimates in Sections 3 and 4 are detailed and internally coherent; in particular, Lemmas 2.4, 3.2, 3.3, 4.1, and 4.2 track the dependence on ε and on the construction-dependent constants in a plausible way, and overlaps are handled carefully through the equivalence classes [w]. The principal reservations concern the dimension-coincidence part: Lemma 5.3 depends on an unreviewed preprint [8], and the counterexample in Theorem 1.4 is asserted without computation. It is worth emphasizing that Theorem 1.1 and Corollary 1.2 do not rely on [8] and appear to stand on their own.

major comments (2)
  1. [§5, Lemma 5.3] The 'if' direction of Lemma 5.3 invokes [8, Theorem 3.1], an unreviewed preprint, to obtain a uniform constant C with C^{-1}#{x_w : w∈E_k, y_w=y'} ≤ #{x_w : w∈E_k, y_w=y} ≤ C#{x_w : w∈E_k, y_w=y'} for all y,y′∈πI_k. This uniform fiber estimate is exactly what makes h_top(π^{-1}(y)) independent of y, and it is therefore load-bearing for the equivalence between dim_H X = dim_B X and Bowen's equality (5.2). Since that implication is used in Theorem 5.1, in the final clause of Theorem 1.3 ('if dim_B X = dim_A X, then dim_H X = dim_B X = dim_A X'), and in the irreducible case of Theorem 1.4, the coincidence results are conditional on a result that is neither stated nor proved in the paper, and whose hypotheses in [8] are not checked against the graph-directed shifts of finite type arising from the irreducible components H_i. Please state the needed uniform fiber estimate as a lemma and prove it, or give a self-contained proof for the specific shifts considered here.
  2. [§5, proof of Theorem 1.4] The counterexample in the final paragraph of Section 5 is asserted rather than verified: after referring to Figure 6, the paper states 'It is not hard to check that dim_B π(X_a)=dim_B π(X_b)=1, and dim_A X=2, dim_L X=1, dim_B X=dim_H X=3/2' and gives no further computation. This example is the sole evidence for the claim that the dichotomy fails when G is not irreducible, so the claimed dimension values are load-bearing. Please supply the verification, for instance by computing α and β from Theorem 1.1 and using the Fraser–Jurga formula (1.2), or by giving the adjacency matrices and digit assignments in enough detail to make the calculation routine.
minor comments (4)
  1. [§4, Lemma 4.2] The sentence 'Choose a k ≥ 1 such that k = ⌊k log_n m⌋ + s_j + h' is confusing as written; the intended condition is k − ⌊k log_n m⌋ = s_j+h, and the existence of such k for each prescribed value s_j+h should be stated explicitly.
  2. [§2.1] The notation for box dimensions is not always visually distinguishable: the paper uses both 'dim_B' and 'dimB' in the same paragraphs, with underlining or overlining that is easy to lose. Please use unambiguous symbols throughout, especially in Lemma 3.1 and in the statements of Theorems 1.3 and 1.4.
  3. [§2.2, definition of β_k] The definition of β_k is intricate: the roles of w, y, and y′ are hard to parse, and the indicator factor in the inner minimum is redundant with the positivity condition in the set over which the minimum is taken. A short explanatory paragraph after (2.8) would improve readability.
  4. [§5, Lemma 5.2] In the construction of the infinite word ˙y, the proof writes 'for all k in N' where it must mean 'for all sufficiently large k' after choosing N; the concatenation argument would also benefit from an explicit sentence explaining why the connecting words of length at most S do not change the growth rate.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Assouad and lower dimension formulas are derived from independent graph-counting and projection box-dimension inputs, with no fitted parameter renamed as a prediction.

full rationale

The central quantities α and β defined in Section 2.2 are built from dim_B π(X_v) of the graph-directed self-similar projection system and from counts of admissible words; these are external inputs, not the output dimensions. The approximate-square estimates in Lemmas 2.4, 3.2, 3.3, 4.1 and 4.2 connect these inputs to the Assouad and lower dimensions directly, and the proof of Theorem 1.1 uses only these estimates. No equation in the paper reduces to the target by definition, and no fitted parameter is later renamed as a prediction. Theorems 1.3 and 1.4 use Bowen's inequality and external theorems (Fraser–Jurga, Ledrappier–Walters, and Feng's preprint); these are cited results rather than self-citations by the present authors, and none of them is equivalent to the paper's own conclusion. The 'if' direction of Lemma 5.3 does rely on [8, Theorem 3.1], an unreviewed preprint, which is a genuine correctness or completeness risk, but it is not circularity: Zhou Feng is not an author of this paper, and the uniform fiber estimate is not derived from the paper's definitions or outputs. Similarly, the asserted example in Theorem 1.4, with dim_A X = 2, dim_L X = 1, and dim_B X = dim_H X = 3/2, is stated without a written verification; that is a support concern rather than a circular step. The derivation chain is therefore self-contained with respect to its own conclusions.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new entities and no fitted constants. It relies on a small set of external theorems about dimension and entropy, one of which is an unreviewed preprint. The quantities α and β are defined from the graph data and box dimensions of projections, and are not free parameters.

assumptions (6)
  • standard math Das-Ngai Theorems 1.1 and 2.7: graph-directed self-similar IFS with finite type overlaps has equal Hausdorff and box dimension for the union of attractors.
    Invoked in Proposition 2.1 to justify that dim_B π(X_v) exists and equals dim_H π(X_v).
  • standard math Ledrappier-Walters relative variational principle, equation (5.6).
    Used in Lemma 5.3 to relate h_top(I) to an integral of fiber entropies.
  • standard math Bowen's inequality h_top(I) ≤ h_top(πI) + sup_y h_top(π^(-1)(y)).
    Used in Theorem 1.3 and Lemma 5.3 to bound dim_B by dim_A.
  • standard math Fraser-Jurga formula (1.2) for the box dimension of X in terms of irreducible components.
    Used in the proof of Theorem 1.3 and Theorem 1.4.
  • domain assumption Feng's [8, Theorem 3.1]: uniform fiber count bounds for irreducible subshifts.
    Used in the 'if' direction of Lemma 5.3; the cited result is an unreviewed preprint and is not proved in this manuscript.
  • standard math Fraser-Jurga Corollary 3.2: dim_H X=dim_B X iff there is a measure of maximal entropy on I projecting to a measure of maximal entropy on πI.
    Used in the 'only if' direction of Lemma 5.3.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Assouad and lower dimensions of graph-directed Bedford-McMullen carpets." pith.science (2026). https://pith.science/paper/3SVDJSQU

@misc{pith2026241115407,
  author       = {Pith},
  title        = {Pith review of: Assouad and lower dimensions of graph-directed Bedford-McMullen carpets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3SVDJSQU}},
  note         = {Machine review of arXiv:2411.15407}
}
read the original abstract

We calculate the Assouad and lower dimensions of graph-directed Bedford-McMullen carpets, which reflect the extreme local scaling laws of the sets, in contrasting with known results on Hausdorff and box dimensions. We also investigate the relationship between distinct dimensions. In particular, we identify an equivalent condition when the box and Assouad dimension coincide, and show that under this condition, the Hausdorff dimension attains the same value.

Figures

Figures reproduced from arXiv: 2411.15407 by the authors.

Figure 1
Figure 1. The location of Qls (p, q) (resp. Qks ) in ψw([0, 1]2 ) (resp. ψw˜([0, 1]2 )). = X v ′∈t(v,[w]) X [w′ ]∈[El(k)−l ′(k)]:i(w′)=v′ , ψw◦ψw′ ((0,1)2)⊆Q◦ l(k) (p,q) Nk−l ′(k) [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Covering Q◦ k (p, q) with approximate squares of level k ′ in Cases (i), (ii). Case (i): l ′ < k. In this case, by firstly dividing Qk(p, q) into rectangles in terms of ψw˜([0, 1]2 ) ∩ Qk(p, q) with ˜w ∈ El ′ , then covering each rectangle with approximate squares of level k ′ (see [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. The choice of Qk(p, q) according to s, v, v˜ and y. = X [w′ ]∈[Es]: i(w′)=v,yw′ =y Nk ′−l [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Covering Q◦ k (p, q) with approximate squares of level k ′ . Case (i): l ′ < k. This case is similar to Case (i) in Lemma 3.2. By (3.5), we have (see [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]
Figure 5
Figure 5. Figure 5: The choice of Qk(p, q) according to j, y and y ′ . Choose a k ≥ 1 such that k = ⌊k logn m⌋ + sj + h and let l = ⌊k logn m⌋. Since |w| ≥ #V and {i(w1), · · · , i(w|w| ), t(w)} ⊆ V , there must exist ˜w ∈ El such that t( ˜w) ∈ t([w]). Let p = Pl i=1 xw˜in l−i and q = Pl …
Figure 6
Figure 6. Figure 6: An example of graph-directed Bedford-McMullen carpet family. At this time, n = 4, m = 3. It is not hard to check that dimB π(Xa) = dimB π(Xb) = 1, and for X = Xa ∪ Xb, dimA X = 2 and dimL X = 1, dimB X = dimH X = 3 2 . □ Acknowledgments The authors are grateful to Dr. …

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 17 canonical work pages

  1. [8]

    Feng, On the coincidence of the Hausdorff and box dimensions for some affine-invariant sets, arXiv: 2405.03213

    Z. Feng, On the coincidence of the Hausdorff and box dimensions for some affine-invariant sets, arXiv: 2405.03213

    arXiv

  2. [1]

    Bara´ nski, Hausdorff dimension of the limit sets of some planar geometric constructions, Adv

    K. Bara´ nski, Hausdorff dimension of the limit sets of some planar geometric constructions, Adv. Math. 210 (2007), no. 1, 215–245

    work page 2007

  3. [2]

    Bedford, Crinkly curves, Markov partitions and box dimensions in self-similar sets, Ph.D

    T. Bedford, Crinkly curves, Markov partitions and box dimensions in self-similar sets, Ph.D. Thesis, University of Warwick, 1984

    work page 1984

  4. [3]

    Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans

    R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971), 401–414

    work page 1971

  5. [4]

    Das and S.-M

    M. Das and S.-M. Ngai, Graph-directed iterated function systems with overlaps, Indiana Univ. Math. J. 53 (2004), no. 1, 109–134

    work page 2004

  6. [5]

    T. Das, D. Simmons, The Hausdorff and dynamical dimensions of self-affine sponges: a dimension gap result, Invent. math. 210 (2017), 85–134

    2017

  7. [6]

    Falconer, Fractal geometry, Mathematical foundations and applications, Third edition, John Wiley & Sons, Ltd., Chichester, 2014

    K. Falconer, Fractal geometry, Mathematical foundations and applications, Third edition, John Wiley & Sons, Ltd., Chichester, 2014. xxx+368 pp

    work page 2014

  8. [7]

    Feng and Y

    D.J. Feng and Y. Wang, A class of self-affine sets and self-affine measures, J. Fourier Anal. Appl. 11 (2005), no. 1, 107–124

    work page 2005

Show all 22 references
  1. [9]

    Fraser, On the packing dimension of box-like self-affine sets in the plane, Nonlinearity, 25 (2012), no

    J.M. Fraser, On the packing dimension of box-like self-affine sets in the plane, Nonlinearity, 25 (2012), no. 7, 2075–2092

    2012

  2. [10]

    Fraser, Assouad type dimensions and homogeneity of fractals, Trans

    J.M. Fraser, Assouad type dimensions and homogeneity of fractals, Trans. Amer. Math. Soc. 366 (2014), no. 12, 6687–6733

    2014

  3. [11]

    Fraser, On the Lq-spectrum of planar self-affine measures, Trans

    J.M. Fraser, On the Lq-spectrum of planar self-affine measures, Trans. Amer. Math. Soc. 368 (2016), no. 8, 5579–5620

    2016

  4. [12]

    Fraser, Assouad dimension and fractal geometry, Cambridge Tracts in Math., 222 Cambridge Uni- versity Press, Cambridge, 2021

    J.M. Fraser, Assouad dimension and fractal geometry, Cambridge Tracts in Math., 222 Cambridge Uni- versity Press, Cambridge, 2021

    2021

  5. [13]

    Fraser and N

    J.M. Fraser and N. Jurga, The box dimensions of exceptional self-affine sets in R3, Adv. Math. 385 (2021), No. 107734

    2021

  6. [14]

    Fraser and N

    J.M. Fraser and N. Jurga, Box dimensions of (×m, ×n)-invariant sets, Indiana Univ. Math. J. 72 (2023), no. 6, 2341–2367. 24 HUA QIU, QI W ANG, AND SHUF ANG W ANG ∗

    2023

  7. [15]

    Kenyon and Y

    R. Kenyon and Y. Peres, Hausdorff dimensions of sofic affine-invariant sets, Israel J. Math. 94 (1996), 157–178

    1996

  8. [16]

    Kolossv´ ary,The Lq spectrum of self-affine measures on sponges, J

    I. Kolossv´ ary,The Lq spectrum of self-affine measures on sponges, J. London Math. Soc., 108 (2023), 666-701

    2023

  9. [17]

    Lalley and D

    S.P. Lalley and D. Gatzouras, Hausdorff and box dimensions of certain self-affine fractals, Indiana Univ. Math. J. 41 (1992), no. 2, 533–568

    1992

  10. [18]

    Larman, A new theory of dimension, Proc

    D.G. Larman, A new theory of dimension, Proc. London Math. Soc. (3) 17 (1967), 178– 192

    1967

  11. [19]

    Ledrappier and P

    F. Ledrappier and P. Walters, A relativised variational principle for continuous transformations, J. London Math. Soc. (2) 16 (1977), no. 3, 568–576

    1977

  12. [20]

    Mackay, Assouad dimension of self-affine carpets, Conform

    J.M. Mackay, Assouad dimension of self-affine carpets, Conform. Geom. Dyn. 15 (2011), 177–187

    2011

  13. [21]

    McMullen, The Hausdorff dimension of general Sierpi´ nski carpets,Nagoya Math

    C. McMullen, The Hausdorff dimension of general Sierpi´ nski carpets,Nagoya Math. J. 96 (1984), 1–9

    1984

  14. [22]

    Olsen, Self-affine multifractal Sierpinski sponges in Rd, Pacific J

    L. Olsen, Self-affine multifractal Sierpinski sponges in Rd, Pacific J. Math., 183 (1998), no. 1, 143-199. School of Mathematics, Nanjing University, Nanjing, 210093, P. R. China. Email address : huaqiu@nju.edu.cn School of Mathematics, Nanjing University, Nanjing, 210093, P. ...

    1998

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.