{"id":"1a59e6f0-68b6-4032-847e-a78f2c26786f","arxiv_id":"2412.02278","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For Bedford-McMullen sponge systems with r coordinates, the metric mean dimension is a weighted sum of topological entropies of projections, and the mean Hausdorff dimension equals a weighted topological entropy divided by log m_1.","lead":"This paper proves explicit formulas for the metric mean dimension and mean Hausdorff dimension of Bedford-McMullen sponge systems, infinite-dimensional self-affine sets built from subshifts over finite alphabets with r at least 3 base scales. The formulas express these dynamical dimensions through topological entropy of projections and weighted topological entropy, generalizing Tsukamoto's results for planar carpet systems.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.12 is false as stated: the l≥2rN+1 pigeonhole does not force a separated pair of approximate cubes, so the lower bound for mean Hausdorff dimension is not proved as written.","rationale":"Reading in good faith, the theorem is a natural extension of Tsukamoto's carpet results and most of the machinery is standard: the metric mean dimension computation is parameter-free and the upper bound for the mean Hausdorff dimension follows the expected covering/measure route. The single load-bearing defect is the combinatorial control in Lemma 3.12. The reader flagged this as the least verifiable step; it is in fact false as stated. The failure is not merely a missing factor of two in a limit that would wash out; it breaks the bound |C_j|≤2rN that converts a covering of X_{Ω|N} into a lower bound on Hausdorff dimension. However, the quantitative repair is straightforward: with l≥2^{rN}+1 the pigeonhole argument is valid, and the subsequent chain of inequalities still closes if the ε-smallness condition is changed from 2rm_1^s ε^{δ/2}<1/2 to 2^r m_1^s ε^{δ/2}<1/2. Thus the central formula is plausibly correct and the manuscript should be accepted only after the lemma and the constants in §3.4.2 are corrected. The one-sided shift issue is real but secondary: all invariants used are defined through forward Bowen metrics and pass to the natural extension unchanged, so it does not affect the numerical content of Theorem 1.1.","tokens_in":25580,"tokens_out":22515,"duration_ms":237912,"concrete_test":"Run an exhaustive check of the counterexample: set r=3, N=2, m=(2,3,4), M=2, and Ω the full shift. For each coordinate q construct the four L_q-digit blocks realizing partial sums (0,0), (δ_q,0), (0,δ_q), (δ_q,δ_q) with δ_q=m_q^{-L_q}; form all 64 triples and pick any 13. Verify that for every pair and every q the l∞ norm of the difference of the two partial-sum vectors is <2δ_q, and that the corresponding approximate cubes have l∞ distance <m_1^{-M}=1/4. If confirmed, Lemma 3.12 is false. Then re-derive the lower-bound chain with |C_j|≤2^{rN} and the modified ε-condition 2^r m_1^s ε^{δ/2}<1/2; check that (3.11) and the final inequality D(E_j)^{log_{m_1}Z_N-Nδ}>1 still close.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing concern is the combinatorial Lemma 3.12 used in the lower bound for mean Hausdorff dimension (§3.4.2). The proof asserts that from l≥2rN+1 distinct approximate cubes one can find two whose partial sums differ by at least 2m_q^{-L_q(M)} in some coordinate q. This is not a valid pigeonhole. If for every q and every time coordinate the partial-sum values lie in an interval of length <2δ_q, each coordinate can still take two grid values, allowing up to 2^{rN} distinct cubes with no pair satisfying (3.9); 2rN+1 is far too small. Concrete counterexample: r=3, N=2, m=(2,3,4), M=2, full shift Ω, so L=(2,1,1), δ=(1/4,1/3,1/4). For each q take the four vectors (0,0), (δ_q,0), (0,δ_q), (δ_q,δ_q) over the two time coordinates and choose 13 distinct triples. Any two differ by at most δ_q in each q, never 2δ_q, and their approximate cubes touch, so no pair is m_1^{-M}-separated. This contradicts Lemma 3.12 and propagates to the bound |C_j|≤2rN in the proof of (3.12). The argument is repairable by replacing the lemma with l≥2^{rN}+1 and adjusting (3.10) to 2^r m_1^s ε^{δ/2}<1/2, but as written the lower-bound proof is invalid.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper extends Tsukamoto's mean-dimension theory for Bedford-McMullen carpet systems to sponge systems of arbitrary dimension r≥3. For a subshift Ω and integer bases m_1≤...≤m_r, the author defines the associated subsystem X_Ω of the r-fold Hilbert cube and proves explicit formulas for the metric mean dimension and the mean Hausdorff dimension in terms of the topological entropies of the projections π_i(Ω) and of Alibabaei's weighted topological entropy. The proof proceeds by covering and separating estimates with approximate cubes, with upper and lower bounds established separately. A condition under which the two invariants coincide is also given.","tokens_in":25862,"tokens_out":20610,"duration_ms":190966,"significance":"If the main result is established, it is a substantial and natural extension of Tsukamoto's r=2 result: it gives closed-form, parameter-free formulas for both invariants for arbitrary r and connects the mean Hausdorff dimension of sponge systems to the weighted topological entropy and its variational principle. The metric mean dimension computation and the upper bound for the mean Hausdorff dimension are carefully executed, and the paper contains no fitted parameters or circular assumptions. However, the lower-bound proof for the mean Hausdorff dimension rests on Lemma 3.12, which is false as stated; the flaw is localized and repairable, so the contribution is likely correct after revision.","major_comments":[{"comment":"Lemma 3.12 is false as stated, and the lower bound (3.12) therefore relies on an invalid pigeonhole argument. For the full shift Ω and any M large enough that L_q(M)≥1 for all q, choose for each q and each time coordinate t=1,...,N the partial sum ∑_{k=1}^{L_q(M)} x_{kq,t} m_q^{-k} to be either 0 or m_q^{-L_q(M)} (put all leading digits 0, or put x_{L_q,q,t}=1 and all earlier leading digits 0). This gives 2^{rN} distinct leading tuples, hence 2^{rN} distinct approximate cubes. For any two of these cubes and every q, the coordinate-wise partial sums differ by at most m_q^{-L_q(M)} in ℓ∞-norm, so inequality (3.9), which requires a difference of at least 2m_q^{-L_q(M)}, fails for every pair. Since 2rN+1≤2^{rN} for r≥3 and N≥1, the assumption of the lemma can be satisfied while its conclusion fails. The lemma is used to conclude |C_j|≤2rN in the proof of (3.12). The repair is to replace the threshold 2rN+1 by 2^{rN}+1; then |C_j|≤2^{rN}, and the existing smallness condition (3.10) still gives the desired factor (2^r m_1^s ε^{δ/2})^N<1/2, yielding ∑ D(E_j)^{log_{m1} Z_N-Nδ}>1. As written, the lower-bound proof is incomplete.","section":"§3.4.2, Lemma 3.12"},{"comment":"Section 2.2 defines a TDS as a pair with T a homeomorphism, but the sponge system (X_Ω,σ) is a one-sided shift, which is not invertible. As written, mdimM(X_Ω,σ,d) and mdimH(X_Ω,σ,d) in Theorem 1.1 are therefore not covered by the definitions. The proofs only use forward Bowen metrics, and the numerical quantities are unaffected by this issue, but the paper should explicitly extend the definitions to non-invertible continuous maps using forward metrics, or state that one passes to the natural extension and that the invariants coincide.","section":"§2.2 and §1.2"}],"minor_comments":[{"comment":"In the proof of Lemma 3.1, 'the first equality holds' should read 'the first inequality holds'; the second displayed assertion is also an inequality, not an equality.","section":"§3.1, Lemma 3.1"},{"comment":"In the display of Lemma 3.6, the index j in u_j(⌊c_i t/L_i⌋) is undefined; it should be u_i.","section":"§3.3, Lemma 3.6"},{"comment":"In the abstract, 'whose metric mean dimension and mean Hausdorff dimension does not coincide' should be 'do not coincide'.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"The main formulas are likely correct and the paper is a solid extension of Tsukamoto's work, but the false Lemma 3.12 blocks the lower-bound proof as written. The repair is localized: the threshold in the lemma should be 2^{rN}+1 instead of 2rN+1, and the subsequent constants go through with the existing condition (3.10). I recommend requesting a revision rather than rejecting, and asking the author to address the non-invertibility gap in the definitions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: the paper gives new explicit formulas for the metric mean dimension and mean Hausdorff dimension of Bedford-McMullen sponge systems for arbitrary r≥3. That's a real advance over Tsukamoto's r=2 carpets. But the lower-bound proof for the mean Hausdorff dimension formula depends on Lemma 3.12, and that lemma is false as stated. A counterexample: r=4, N=1, all m_i=2, full shift, M=1. There are 16 distinct approximate cubes, and all of them contain the point (1/2,...,1/2), so no two are m_1^{-1}-separated; taking any 9 of them satisfies the lemma's hypothesis l≥2rN+1. Consequently the bound |C_j|≤2rN in Section 3.4.2 is unjustified and the lower bound for (1.3) is not proved as written. The stress-test note's own counterexample is off — with m=(2,3,4) and M=2, L_2=1 so there are not two time coordinates to vary — but the criticism is correct: the pigeonhole count in Lemma 3.12 is the wrong one. The repair looks routine: replace 2rN with something like 2^{rN} in the lemma and adjust the constants in (3.10). I expect the theorem to survive, but the manuscript needs that fix.\n\nWhat the paper does well: the formulas are genuinely new and plausible; the metric mean dimension calculation is clean; the upper bound for mean Hausdorff dimension is detailed; and the use of Alibabaei's weighted topological entropy is appropriate. No fitted parameters, no circularity, and the citations to Kenyon-Peres, Tsukamoto, and Alibabaei are the right ones.\n\nMinor issues: the standing TDS definition requires a homeomorphism, while the sponge system is a one-sided shift; the proofs only use forward Bowen metrics, so this is a presentation gap rather than a substantive problem. Lemma 3.1 calls an inequality an equality, and Lemma 3.6 has an undefined index j in the displayed limit.\n\nThe paper is for specialists in mean dimension and fractal dynamics. It deserves a serious referee; after the counting lemma is repaired it should be publishable. I'd send it to review.","headline":"New and likely correct formulas, but the lower bound for mean Hausdorff dimension rests on a false counting lemma (Lemma 3.12) and needs repair.","tokens_in":26415,"tokens_out":17315,"would_cite":true,"duration_ms":162740,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["28A80","37C45","28D20","37B40","37A35"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves exact, closed-form formulas for both the metric mean dimension and the mean Hausdorff dimension of every Bedford-McMullen sponge system with any number $r\\ge 3$ of coordinate directions.","keywords":["metric mean dimension","mean Hausdorff dimension","weighted topological entropy","Bedford-McMullen sponge","infinite-dimensional fractal","subshift entropy","self-affine set"],"falsifier":"Test the cube-counting lemma directly: take $r=3$ with bases $m_1=2,m_2=4,m_3=8$ and $\\Omega$ the full shift, and for small $N,M$ enumerate all distinct approximate cubes $Q_{N,M}(x)$. Lemma 3.12 asserts that among any $2rN+1$ distinct cubes two are at distance at least $m_1^{-M}$; finding a counterexample would break the lower-bound proof of the mean Hausdorff formula. A second test is numerical: for a family of subshifts $\\Omega$, compute the covering numbers of $(X_\\Omega,d_N,\\varepsilon)$ directly and compare the limiting ratios with the two closed formulas in Theorem 1.1.","tokens_in":25325,"feed_emoji":"🧽","tokens_out":15638,"duration_ms":132098,"temperature":0.7,"pith_summary":"The paper proves that the two central quantitative invariants of mean dimension theory—metric mean dimension and mean Hausdorff dimension—can be computed exactly for Bedford-McMullen sponges with any number $r\\ge 3$ of digit coordinates, not just the planar carpet case $r=2$. A sponge is built from a subshift $\\Omega$ of digit sequences in $\\{0,\\dots,m_1-1\\}\\times\\cdots\\times\\{0,\\dots,m_r-1\\}$ by coding each coordinate stream as a base-$m_i$ expansion, producing a shift-invariant set in $([0,1]^r)^{\\mathbb{N}}$. The metric mean dimension is shown to be a weighted sum of the ordinary topological entropies of $\\Omega$ and of its projections onto the first $i$ coordinates, while the mean Hausdorff dimension is shown to equal a weighted topological entropy divided by $\\log m_1$. The result reduces two seemingly complicated infinite-dimensional invariants to data read directly from the subshift and its projections.","feed_headline":"Infinite sponges get exact mean-dimension formulas","feed_subtitle":"Metric mean dimension becomes an entropy sum; mean Hausdorff becomes weighted entropy, for every r.","key_machinery":"The argument is carried by approximate cubes $Q_{N,M}(x_1,\\ldots,x_r)$: for each level $M$, these are the sets of sponge points whose first $L_i(M)=\\lfloor M\\log m_1/\\log m_i\\rfloor$ digits in coordinate $i$ agree with those of a centre point. Each such cube has diameter on the order of $m_1^{-M}$, and the numbers of such cubes are controlled by $|\\Omega|_N^{L_r(M)}\\prod_{i=1}^{r-1}|\\pi_i(\\Omega)|_N^{L_i(M)-L_{i+1}(M)}$, which yields the metric mean dimension formula. For the mean Hausdorff dimension, the load-bearing object is the weighted topological entropy $h^a_{\\mathrm{top}}$ defined by nested covers with weight vector $a_i=\\log m_{r-i}/\\log m_{r-i+1}$; its value is computed through the fibre counts $Z_N$ of Lemma 3.3, and the upper and lower bounds are matched by a product measure on $(\\Omega|_N)^{\\mathbb{N}}$ whose conditional probabilities make the masses of approximate cubes explicit.","core_discovery":"On the paper's own terms, the discovery is that both invariants of an infinite-dimensional sponge system are already determined by entropy data of the coding subshift. For a sponge system $(X_\\Omega,\\sigma,d)$ with bases $2\\le m_1\\le\\cdots\\le m_r$, Theorem 1.1 gives $\\operatorname{mdim}_M(X_\\Omega,\\sigma,d)=h_{\\mathrm{top}}(\\Omega,\\sigma)/\\log m_r+\\sum_{i=1}^{r-1}(1/\\log m_i-1/\\log m_{i+1})h_{\\mathrm{top}}(\\pi_i(\\Omega),\\sigma)$ and $\\operatorname{mdim}_H(X_\\Omega,\\sigma,d)=h^a_{\\mathrm{top}}(\\{(\\pi_i(\\Omega),\\sigma)\\}_{i=1}^r,\\{\\tau_i\\}_{i=1}^{r-1})/\\log m_1$, where the weights are $a_i=\\log m_{r-i}/\\log m_{r-i+1}$, $\\pi_i$ drops the last $r-i$ coordinates, and $\\tau_i$ are the corresponding factor maps. In particular, the paper establishes for arbitrary $r$ that metric mean dimension is a linear combination of standard entropies, while mean Hausdorff dimension is governed by the weighted topological entropy of the whole projection tower. It also gives a uniform fibre-growth condition under which the two invariants coincide.","pith_inferences":["A testable extension the paper leaves implicit is that the same approximate-cube counts should compute metric mean dimension for graph-directed or sofic variants of sponges, provided the analogue of the fibre counts $Z_N$ is computable.","Passing to the natural extension of the one-sided shift would likely repair the homeomorphism gap without changing the formulas, because topological entropy and the projection entropies are invariant under this passage; the paper does not discuss this.","Equality of the two mean dimensions might hold under weaker fluctuation bounds on the fibre counts than the uniform growth condition (3.3); a numerical search over subshifts with slowly varying fibre sizes could probe how sharp that condition is."],"forward_implications":["When $r=2$, formulas (1.2) and (1.3) reduce to the known carpet-system result, so the sponge formulas contain the earlier planar theory as the first case.","If $\\Omega$ has uniformly growing word complexity in the sense of (3.3), then the mean Hausdorff dimension equals the metric mean dimension; this is the dynamical analogue of the uniform-fibres case where Hausdorff and Minkowski dimensions of sponges agree.","The variational form of the two formulas shows that the gap between mean Hausdorff dimension and metric mean dimension is exactly the gap between maximizing all projected entropies with a single measure and maximizing each projection separately.","Since $h_{\\mathrm{top}}(\\pi_i(\\Omega),\\sigma)\\le h_{\\mathrm{top}}(\\Omega,\\sigma)$ and the coefficients $1/\\log m_i-1/\\log m_{i+1}$ are nonnegative, the metric mean dimension lies between $h_{\\mathrm{top}}(\\Omega,\\sigma)/\\log m_r$ and $h_{\\mathrm{top}}(\\Omega,\\sigma)/\\log m_1$."],"supporting_citations":[{"why":"the carpet-system theorem and the upper/lower bound strategy, including Lemmas 3.5 and 3.7, that this paper extends to r≥3.","marker":"[25]"},{"why":"introduced Bedford-McMullen sponges, supplies the combinatorial Lemma 3.6, and gives the dimension formulas whose dynamical analogues Theorem 1.1 proves.","marker":"[14]"},{"why":"provides the definition of weighted topological entropy for a sequence of systems and the variational principle used to convert the mean Hausdorff formula into measure-theoretic form.","marker":"[1]"},{"why":"established the weighted-entropy equality in the carpet case and the associated Hausdorff dimension computation that the present result generalizes.","marker":"[26]"},{"why":"introduced metric mean dimension as the entropy-resolution growth rate, the quantity formula (1.2) computes.","marker":"[20]"},{"why":"introduced mean Hausdorff dimension and its role as a finer invariant, the quantity formula (1.3) computes.","marker":"[19]"}],"fun_headline_variants":["Sponge mean dimension reduces to entropy sums","Infinite sponges: mean dimension is entropy combination","Metric and Hausdorff sponge dimensions from entropy data","Sponge invariants become weighted entropy formulas","Entropy sums determine infinite sponge mean dimensions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper defines mean dimension for invertible dynamics only, while the sponge system is a one-sided shift that cannot be run backwards; the theorem presupposes that the same definitions and estimates extend to this case without saying so.","fun_headline_variants_meta":{"raw":{"variants":["Sponge mean dimension reduces to entropy sums","Infinite sponges: mean dimension is entropy combination","Metric and Hausdorff sponge dimensions from entropy data","Sponge invariants become weighted entropy formulas","Entropy sums determine infinite sponge mean dimensions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000282,"raw_usage":{"total_tokens":1697,"prompt_tokens":1002,"completion_tokens":695,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":618,"completion_tokens_details":{"reasoning_tokens":625}},"tokens_in":618,"tokens_out":695,"duration_ms":6884,"temperature":1.0,"reasoning_tokens":625,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T23:41:17.917451+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Test the cube-counting lemma directly: take $r=3$ with bases $m_1=2,m_2=4,m_3=8$ and $\\Omega$ the full shift, and for small $N,M$ enumerate all distinct approximate cubes $Q_{N,M}(x)$. Lemma 3.12 asserts that among any $2rN+1$ distinct cubes two are at distance at least $m_1^{-M}$; finding a counterexample would break the lower-bound proof of the mean Hausdorff formula. A second test is numerical: for a family of subshifts $\\Omega$, compute the covering numbers of $(X_\\Omega,d_N,\\varepsilon)$ directly and compare the limiting ratios with the two closed formulas in Theorem 1.1.","supporting_citations":[{"cited_title":"Mean Hausdorff dimension of some infinite dimensional fractals","cited_arxiv_id":"2209.00512","evidence_quote":"the carpet-system theorem and the upper/lower bound strategy, including Lemmas 3.5 and 3.7, that this paper extends to r≥3."},{"cited_title":"Kenyon and Yuval Peres","cited_arxiv_id":null,"evidence_quote":"introduced Bedford-McMullen sponges, supplies the combinatorial Lemma 3.6, and gives the dimension formulas whose dynamical analogues Theorem 1.1 proves."},{"cited_title":"Weighted topological pressure revisited","cited_arxiv_id":"2307.16772","evidence_quote":"provides the definition of weighted topological entropy for a sequence of systems and the variational principle used to convert the mean Hausdorff formula into measure-theoretic form."},{"cited_title":"New approach to weighted topologica l entropy and pressure","cited_arxiv_id":null,"evidence_quote":"established the weighted-entropy equality in the carpet case and the associated Hausdorff dimension computation that the present result generalizes."},{"cited_title":"Mean topologic al dimension","cited_arxiv_id":null,"evidence_quote":"introduced metric mean dimension as the entropy-resolution growth rate, the quantity formula (1.2) computes."},{"cited_title":"Double varia tional principle for mean dimension","cited_arxiv_id":null,"evidence_quote":"introduced mean Hausdorff dimension and its role as a finer invariant, the quantity formula (1.3) computes."}],"review_version":1}