{"id":"2efb68d7-3813-4e7e-9a66-7467e5e023b0","arxiv_id":"2411.15407","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For graph-directed Bedford-McMullen carpets, the Assouad dimension equals log α over log n and the lower dimension equals log β over log n, for growth rates α and β of counting sequences.","lead":"This paper derives formulas for the Assouad and lower dimensions, two extreme measures of local complexity, for graph-directed Bedford-McMullen carpets, a family of self-affine fractals built from a directed graph. The formulas are given in terms of counting sequences α and β, and the paper also identifies exactly when the box and Assouad dimensions coincide.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Assouad/lower dimension formulas are internally coherent; the load-bearing weakness is the box-Assouad coincidence theorem, which depends on an unverified uniform fiber estimate from an unreviewed preprint.","rationale":"I read the paper in good faith and focused on the strongest claim, Theorem 1.1. The construction of alpha and beta in Section 2.2 is elaborate, but the surrounding proofs are detailed and the scaling arguments are consistent. Lemma 2.4 provides the correct counting estimates for projected pieces; Lemma 3.1 establishes tau <= alpha via box dimension; Lemmas 3.2 and 3.3 supply matching upper and lower bounds for Assouad dimension; Lemmas 4.1 and 4.2 do the same for lower dimension using beta. The dropping of the rectangular open set condition is handled by counting equivalence classes of words with the same affine contraction, and I found no place where partial overlaps break the counting. The central formulas therefore appear sound. The genuine risk is external: Lemma 5.3's proof that dim_H X = dim_B X implies equality in Bowen's inequality depends on [8, Theorem 3.1], an unreviewed preprint result. This is not needed for Theorem 1.1, but it is load-bearing for Theorem 1.3 and the first part of Theorem 1.4, which are central to the paper's second stated aim. The reader's conditional verdict is exactly right: the main formulas should be accepted, while the coincidence theorem should be conditional on independent verification of the uniform fiber estimate. I agree with the reader's identification of the weakest assumption.","tokens_in":20187,"tokens_out":31741,"duration_ms":318428,"concrete_test":"Obtain arXiv:2405.03213 and check Theorem 3.1 against the graph-directed shift I_H_i used in Lemma 5.3: for a small irreducible example (e.g., a full shift on two letters with a nontrivial projection), enumerate all admissible words up to length 12 and compute max_y #fiber_k(y) / min_y #fiber_k(y); if this ratio grows with k, the uniform fiber estimate fails and Lemma 5.3 collapses. Independently recompute the dimensions in the Figure 6 example (n=4, m=3) by direct counting of level-k cylinders to confirm that dim_B X = dim_H X = 3/2 while dim_A X = 2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.1 in Sections 3 and 4 is careful and internally consistent: the approximate-square estimates in Lemmas 2.4, 3.2, 3.3, 4.1, and 4.2 track the resolutions correctly, and overlaps are handled through the equivalence classes [w]. I do not see a defect in the central dimension formulas themselves. The load-bearing weakness is in the dimension-coincidence part. Theorem 1.3 and the first part of Theorem 1.4 rely on Lemma 5.3, whose 'if' direction invokes [8, Theorem 3.1]. That theorem asserts that for an irreducible subshift satisfying the weak specification property, the number of admissible words above a given projection word is comparable for all projection words of the same length, with a constant independent of the word and level. This uniform fiber estimate is what makes h_top(pi^{-1}(y)) independent of y, so that dim_H X = dim_B X forces equality in Bowen's inequality (5.1). If [8, Theorem 3.1] is false, or if its hypotheses do not cover the graph-directed shifts of finite type arising from the irreducible components H_i, then the equivalence between dim_B X = dim_A X and dim_H X = dim_B X in Theorem 1.3 and Theorem 1.4 loses its proof. The statement that the example in Theorem 1.4 satisfies dim_A X = 2, dim_L X = 1, and dim_B X = dim_H X = 3/2 is also asserted without a verification, so it cannot independently support the no-dichotomy conclusion.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":26,"tokens_out":16680,"duration_ms":755546,"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":[{"comment":"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.","section":"§5, Lemma 5.3"},{"comment":"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.","section":"§5, proof of Theorem 1.4"}],"minor_comments":[{"comment":"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.","section":"§4, Lemma 4.2"},{"comment":"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.","section":"§2.1"},{"comment":"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.","section":"§2.2, definition of β_k"},{"comment":"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.","section":"§5, Lemma 5.2"}],"recommendation":"major_revision","confidential_remarks":"The central dimension formulas in Sections 3 and 4 appear sound, and the paper is a strong contribution if the two gaps I identify are closed. The main risk is the dependence on the unreviewed preprint arXiv:2405.03213 for Lemma 5.3; you may wish to seek input from an expert on Feng's preprint or require the authors to prove the needed uniform fiber estimate. The example in Theorem 1.4 is likely correct but should be verified explicitly, as it is the basis for the no-dichotomy claim."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me give you a quick read. The paper's central result is the exact Assouad and lower dimension formulas for graph-directed Bedford-McMullen carpets, and that part is in good shape. The formulas in Theorem 1.1 are new, they drop the rectangular open set condition, and they extend Mackay and Fraser to the graph-directed setting. The proofs in Sections 3 and 4 are careful and internally consistent; the approximate-square estimates track the resolutions correctly, and the equivalence classes [w] handle overlaps. I did not find an algebraic error in the main argument.\n\nThe soft spot is the dimension-coincidence part. Theorem 1.3 and the first half of Theorem 1.4 depend on Lemma 5.3, and the 'if' direction of that lemma uses a uniform fiber estimate from Feng's preprint [8] (arXiv:2405.03213). The claim that the number of admissible words above a fixed projection word is comparable for all words is exactly what makes the slice entropies independent of y, and without it the equivalence between dim_B = dim_A and dim_H = dim_B loses its proof. That is a real load-bearing dependency, and the authors acknowledge it. It is not fatal to the main formulas, which do not need Lemma 5.3, but a referee should ask for either an independent proof or a careful verification that Feng's hypotheses apply to the graph-directed SFTs arising here.\n\nThe second gap is smaller but still real: the counterexample in Theorem 1.4 is asserted with 'it is not hard to check' and no computation. The claimed dimensions (dim_A=2, dim_L=1, dim_B=dim_H=3/2) are plausible, but as written the example cannot independently support the no-dichotomy conclusion. That should be expanded in revision.\n\nOverall, the paper is honest and the main result looks right. It deserves a serious referee, not a desk reject. The referee's task is to verify the preprint dependency and the example, not to redo the whole proof. If those two points are resolved, this is a solid contribution to the fractal dimensions literature.","headline":"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.","tokens_in":21039,"tokens_out":2255,"would_cite":true,"duration_ms":19175,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["28A80","28A78"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["Assouad dimension","lower dimension","directed graph","self-affine set","Bedford-McMullen carpet","graph-directed iterated function system","box dimension","Hausdorff dimension"],"falsifier":"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.","tokens_in":19987,"feed_emoji":"📐","tokens_out":15422,"duration_ms":126643,"temperature":0.7,"pith_summary":"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.","feed_headline":"Exact Assouad and lower dimension formulas for directed carpets","feed_subtitle":"Assouad dimension measures thickest local scaling, lower dimension the thinnest; the paper gives exact formulas for both.","key_machinery":"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).","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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 $\\dim_A X=2$."],"supporting_citations":[{"why":"Supplies the box-dimension formula (1.2) for graph-directed carpets that the comparison and coincidence criteria build on.","marker":"[14]"},{"why":"Provides the Hausdorff and box-dimension theory for (×m,×n)-invariant sets that the paper complements with Assouad and lower dimensions.","marker":"[15]"},{"why":"Shows the projection family satisfies a finite-type overlap condition, used to prove the box dimensions of the projections exist.","marker":"[4]"},{"why":"Gives the uniform fiber-counting estimate used in the 'if' direction of Lemma 5.3, making fiber entropies independent of the vertical word.","marker":"[8]"},{"why":"Establishes the first Assouad-dimension formula for self-affine carpets, which the graph-directed result extends.","marker":"[20]"},{"why":"Determines Assouad and lower dimensions for the more flexible box-like carpet class and sets up the twin-dimensional approach used here.","marker":"[10]"},{"why":"Provides the relative variational principle that converts equality of dimensions into an equality of topological entropies.","marker":"[19]"},{"why":"Supplies the entropy inequality (5.1) whose equality case forms the coincidence criterion.","marker":"[3]"}],"fun_headline_variants":["Exact scaling dimensions of directed carpets","Assouad and lower dimensions computed for directed carpets","Directed carpets: exact Assouad and lower dimension formulas","Extreme local scaling rates pinned down for directed carpets","Exact formulas for extreme dimensions of directed carpets"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Exact scaling dimensions of directed carpets","Assouad and lower dimensions computed for directed carpets","Directed carpets: exact Assouad and lower dimension formulas","Extreme local scaling rates pinned down for directed carpets","Exact formulas for extreme dimensions of directed carpets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.002143,"raw_usage":{"total_tokens":8314,"prompt_tokens":946,"completion_tokens":7368,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":562,"completion_tokens_details":{"reasoning_tokens":7309}},"tokens_in":562,"tokens_out":7368,"duration_ms":46799,"temperature":1.0,"reasoning_tokens":7309,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:22:11.298799+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Fraser and N","cited_arxiv_id":null,"evidence_quote":"Supplies the box-dimension formula (1.2) for graph-directed carpets that the comparison and coincidence criteria build on."},{"cited_title":"Kenyon and Y","cited_arxiv_id":null,"evidence_quote":"Provides the Hausdorff and box-dimension theory for (×m,×n)-invariant sets that the paper complements with Assouad and lower dimensions."},{"cited_title":"Das and S.-M","cited_arxiv_id":null,"evidence_quote":"Shows the projection family satisfies a finite-type overlap condition, used to prove the box dimensions of the projections exist."},{"cited_title":"Fraser, Assouad type dimensions and homogeneity of fractals, Trans","cited_arxiv_id":null,"evidence_quote":"Determines Assouad and lower dimensions for the more flexible box-like carpet class and sets up the twin-dimensional approach used here."},{"cited_title":"Ledrappier and P","cited_arxiv_id":null,"evidence_quote":"Provides the relative variational principle that converts equality of dimensions into an equality of topological entropies."},{"cited_title":"Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans","cited_arxiv_id":null,"evidence_quote":"Supplies the entropy inequality (5.1) whose equality case forms the coincidence criterion."}],"review_version":1}