{"id":"7a359969-6577-4227-b3c2-478cfcd218c7","arxiv_id":"2411.19712","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Dynamic asymptotic dimension growth is introduced and shown to be equivalent to asymptotic dimension growth for coarse groupoids, implying amenability for groupoids with sublinear dynamic dimension growth.","lead":"This paper defines a growth version of dynamic asymptotic dimension for group actions and etale groupoids, quantifying how a dimension measure can grow with scale even when it is infinite. It proves that this new invariant matches classical asymptotic dimension growth for coarse groupoids and yields amenability under sublinear growth conditions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.16's proof that dad_{G(X)} ≼ |ad_X is invalid: closures of bounded subsets of X need not cover the Stone–Čech unit space βX.","rationale":"The reader's weakest assumption concerns Proposition 5.1, specifically a missing division by N(R) in inequality (5.1). That is a genuine typo, but it is readily repaired: for any f≼x^α, the corrected inequality follows by choosing c large enough, since f(N(R)R+1)/N(R) = O(c^{α-1}) plus lower-order terms, which tends to 0 as c→∞. The paper only sketches this for the special case f(x)=ceil(x^α), but the general argument is a routine exercise. Thus Proposition 5.1 is not the principal barrier. The more serious issue is in the proof of Theorem 4.16, the central equivalence between asymptotic dimension growth and dynamic asymptotic dimension growth of the coarse groupoid. The second half of the proof constructs an open cover of βX from a bounded cover of X by taking unions of closures of the individual bounded sets. This construction does not cover the corona of βX: finite bounded sets have closures equal to themselves in βX, so the union of their closures is just the original subset of X. Hence the resulting family is not a cover of βX. This invalidates the proof of dad_{G(X)} ≼ |ad_X, which is a load-bearing step for the main theorem and its corollaries. The theorem itself may be true and repairable by a different argument, but as written the proof has a substantial gap. Since the reader's verdict was CONDITIONAL and our concern also calls for a revised proof, the verdict remains CONDITIONAL.","tokens_in":40159,"tokens_out":23001,"duration_ms":197250,"concrete_test":"Take X = Z with the usual metric and R = 2. Use the cover U of X by singletons, with U_0 = {{2k}: k∈Z} and U_1 = {{2k+1}: k∈Z}. Then \\overline{{n}} = {n} in βZ, so the sets U_0 and U_1 as defined in the proof are the even and odd integers; their union is Z, not βZ. Therefore the claim that {U_0,U_1} covers βX is false, and the proof of dad_{G(X)} ≼ |ad_X fails for this admissible input.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 4.16, second half (Section 4.2), the authors start from a cover U of X that is F-bounded and decomposes into ER-separated families U_i, and define U_i := ⊔_{U∈U_i} \\overline{U} in βX, claiming that {U_i} is a compact open cover of βX because U covers X. This is false. In a bounded-geometry discrete space, each bounded U is finite, so \\overline{U}=U. The union of these finite sets is exactly the subset of X corresponding to the family, and the union over all such sets in the cover is X itself, not βX (unless X is finite). For example, taking X=Z, R=2, and the cover by singletons decomposed into evens and odds, the sets U_0 and U_1 are the even and odd integers, whose union is Z, a proper open subset of βZ. Hence {U_i} is not an open cover of βZ, and the subsequent construction of precompact subgroupoids G_i collapses. The remainder of the proof relies on U_i being compact open and covering the unit space to conclude dad_{G(X)} ≼ |ad_X; with the stated construction this step does not follow.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces the notion of dynamic asymptotic dimension growth for actions of discrete groups on compact spaces and for locally compact étale groupoids, generalizing the finite dynamic asymptotic dimension of Guentner–Willett–Yu. It establishes several equivalent definitions of asymptotic dimension growth, connects asymptotic dimension growth of a bounded-geometry discrete metric space to the dynamic asymptotic dimension growth of its coarse groupoid (Theorem 4.16), derives amenability results (Corollary 4.19 and Theorem 6.4), and uses Bartels–Lück–Reich actions to construct group actions with prescribed growth behavior. The central claimed contributions are the equivalence between coarse and dynamic growth (Theorem 4.16) and the amenability of groupoids with slow dynamic asymptotic dimension growth (Theorem 6.4).","tokens_in":40357,"tokens_out":9279,"duration_ms":78172,"significance":"If the main results hold, the paper provides a quantitative dimension invariant that unifies coarse geometry, dynamical systems, and operator algebras, extending known finite-dimensional theorems to a growth setting. The paper is well structured, and the development of three equivalent definitions of asymptotic dimension growth (Section 2) and the partition-of-unity machinery (Section 5) are potentially useful tools. The authors are also careful to cite and build on prior work (GWY17, ANWZ18, STY02, MW20), and the paper does not introduce fitted parameters or circular reasoning. However, the proof of the central equivalence Theorem 4.16 contains a substantial gap, and the auxiliary inequality in Proposition 5.1 is misstated; both issues must be resolved before the results can be accepted.","major_comments":[{"comment":"The second half of the proof aims to show dad_{G(X)} ≼ |ad_X. It defines U_i := ⊔_{U∈U_i} \\overline{U} in βX and asserts that {U_0, ..., U_{|ad_X(R)|}} is a cover of βX because U covers X. This is false. Since X has bounded geometry, each bounded U is finite, so \\overline{U} = U. The union over a family U_i is then just the subset of X covered by that family; it need not be open in βX, and the union over all i is exactly X, not βX. For example, with X=Z and the cover by singletons decomposed into evens and odds, the two sets are the even and odd integers, whose union is Z, a proper dense open subset of βZ. Consequently, the construction of the precompact subgroupoids G_i does not yield the required open cover of the unit space of G(X), and the inequality dad_{G(X)} ≼ |ad_X is not established. This gap directly affects Corollary 4.19 and the group-action consequences in Section 4.3.","section":"Section 4.2, proof of Theorem 4.16"},{"comment":"The displayed inequality (5.1) omits a division by N(R) in the first term. As written, the left-hand side contains the term 2p(f(N(R)R+1)+1), which, under f ≼ x^α, grows like N(R)^α and cannot be made smaller than ε by increasing the constant c. The estimates in equations (5.6)–(5.8) actually give the bound (2p(f(N(R)R+1)+1) + 2p(f(N(R)R+1)+1)^{1/p+1})/N(R), so the correct version of (5.1) should have the first term divided by N(R). With this correction, the existence of c is plausible because f(N(R)R+1) = O(N(R)^α) and α<1, but the authors do not prove this for arbitrary f in the given growth class; Remark 5.7 only treats the concrete function f(x)=⌈x^α⌉. Since Proposition 5.1 is the key tool for Theorem 6.4, a complete argument for general f ≼ x^α is required.","section":"Section 5, equation (5.1)"}],"minor_comments":[{"comment":"The notation for the coarse asymptotic dimension function is inconsistent: Lemma 2.13 defines it as }ad_X, while the proof of Theorem 4.16 uses |ad_X. Please use one symbol consistently.","section":"Throughout"},{"comment":"There is a typo: 'unifromly' should be 'uniformly'. Also, the expression 'ĆadX' appears to be a misprint for the function defined earlier.","section":"Section 2.1, proof of Proposition 2.5"},{"comment":"The indexing of the cover is written as f(r(R+1)csR+1), which is ambiguous; it should clearly indicate the floor/ceiling notation, e.g., f(⌈(R+1)c⌉ R+1). The same notational issue appears in the proof of Proposition 5.1 and in Theorem 6.4.","section":"Section 5, Proposition 5.1 statement"},{"comment":"The word 'equivaraint' should be 'equivariant'.","section":"Section 3.2, Proposition 3.13"},{"comment":"The word 'metrizible' should be 'metrizable'.","section":"Section 4.3, Theorem 4.22"}],"recommendation":"major_revision","confidential_remarks":"The gap in Theorem 4.16 is serious because the proof of the reverse inequality relies on a false assertion about closures in βX. The statement of the theorem may still be true, but the current proof is incomplete. The issue in Proposition 5.1 is more local and appears fixable. I recommend major revision, with emphasis on repairing the βX argument and supplying the missing estimate in Section 5."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the paper's new notion (DAD growth) is natural and it gets real results, but the proof of the claimed equivalence in Theorem 4.16 has a hole in one direction. The forward direction (ad_X is controlled by dad_{G(X)}) is fine, so Corollary 4.19 (subexponential DAD growth of G(X) implies amenability) still goes through. But the reverse direction, which is part of the headline 'if and only if', does not follow from the argument as written.\n\nWhat is genuinely there: a clean definition of dynamic asymptotic dimension growth for actions and groupoids, a useful bridge to BLR-type conditions, a nice construction (Theorem 4.22) of free minimal Cantor actions for groups like Z wreath Z with polynomial growth, and an amenability theorem (6.3 and 6.4) using a partition of unity. The exposition follows the GWY strategy closely, and the citations to ANWZ, Ozawa, Ma-Wu, etc. are appropriate.\n\nThe soft spots, in order of severity:\n\n1. The step in Theorem 4.16 (Section 4.2, reverse direction) is wrong as written. The authors set U_i = disjoint union over U in U_i of the closure of U and claim this is a compact open cover of beta X. But each U is finite in a bounded-geometry discrete space, so closure of U equals U; the union over the family is just a subset of X. For X = Z with the cover by singletons, the 'cover' of beta Z is Z itself. You need the closures of the whole families, not of individual members, and then the subgroupoid control becomes unclear. This is not a cosmetic typo; it breaks the reverse inequality.\n\n2. Inequality (5.1) in Proposition 5.1: the first term is missing a division by N(R). With the natural correction, the estimate works when f is at most x^alpha (0 < alpha < 1), so this looks fixable.\n\n3. Proposition 3.13 defers to prior work; minor.\n\nThe paper isn't incoherent, and the forward direction plus the amenability results mean it has real value. But as it stands, the main equivalence is not proven. I'd send it to a serious referee; they should either fix the cover argument or downgrade the claim to the direction that actually goes through.","headline":"Interesting new invariant and several good results, but the proof of the main equivalence in Theorem 4.16 has a real gap in one direction.","tokens_in":40932,"tokens_out":13621,"would_cite":false,"duration_ms":112244,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["54F45","22A22","37A55","37B05","46L05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that a bounded-geometry metric space and its coarse groupoid share the same dimension growth, and that subexponential ($x^\\alpha$, $\\alpha<1$) growth forces amenability.","keywords":["dynamic asymptotic dimension growth","group actions","étale groupoids","coarse groupoid","amenability","asymptotic dimension growth","partition of unity","property A"],"falsifier":"Check the displayed inequality (5.1) in Proposition 5.1: as written the left-hand side lacks division by the auxiliary scale $N(R)$. With the natural correction, test whether the constant $c$ asserted to exist in Remark 5.7 actually makes the corrected inequality hold for the model function $f(x)=\\lceil x^{1/2}\\rceil$ at $R=1$ and $\\varepsilon=0.1$; if no such $c$ exists, or if some admissible $f\\le x^\\alpha$ violates the corrected estimate, the proof of Theorem 6.4 does not go through.","tokens_in":39918,"feed_emoji":"📈","tokens_out":11520,"duration_ms":87324,"temperature":0.7,"pith_summary":"This paper introduces a quantitative version of dynamic asymptotic dimension, tracking how the minimal number of cover pieces needed to control arrows of bounded length grows as the scale increases. The central result is an equivalence: for any discrete metric space $X$ of bounded geometry, the asymptotic dimension growth of $X$ is the same growth type as the dynamic asymptotic dimension growth of its coarse groupoid $G(X)$. From this, the paper shows that subexponential dynamic asymptotic dimension growth of $G(X)$ implies that $G(X)$ is amenable. A more general theorem states that every $\\sigma$-compact étale groupoid with compact unit space and dynamic asymptotic dimension growth at most $x^\\alpha$ ($0<\\alpha<1$) is amenable, providing a dimension-based route to amenability for groupoids and many concrete examples.","feed_headline":"Subexponential dimension growth forces amenability","feed_subtitle":"For bounded-geometry spaces and coarse groupoids, dimension growth matches; slow growth yields amenability.","key_machinery":"The central objects are the dynamic asymptotic dimension function $\\mathrm{dad}_G$ of an étale groupoid (the minimal number of open sets in a cover of the unit space such that all arrows of length $<R$ with source and range in one set lie in a precompact subgroupoid) and the coarse groupoid $G(X)$ of a bounded-geometry metric space, whose arrows encode proximity at all scales. The argument's load-bearing identity is $\\mathrm{ad}_X \\asymp \\mathrm{dad}_{G(X)}$ (Theorem 4.16), obtained by translating the coarse-geometric formulation of asymptotic dimension growth into controlled separations inside $G(X)$. For amenability, the key mechanism is the generalized partition of unity of Proposition 5.1: a growth bound $f\\le x^\\alpha$ yields continuous functions $(\\varphi_i)$ on the unit space with $\\sum_i \\varphi_i^p = 1$ and uniformly small variation along arrows of length $<R$, from which an almost invariant measure is assembled; here $p$ is an integer depending only on $\\alpha$.","core_discovery":"The paper's central claim is that the growth type of the dynamic asymptotic dimension function—the minimal $m$ such that for each scale $R$ the arrows of length $<R$ with source and range in a single cover element lie in a precompact subgroupoid—is a meaningful quantitative invariant, not just a finiteness test. The load-bearing equivalence is Theorem 4.16: for a discrete metric space $X$ of bounded geometry with coarse groupoid $G(X)$ and any nondecreasing $f$, $\\mathrm{ad}_X \\asymp f$ if and only if $\\mathrm{dad}_{G(X)} \\asymp f$. A direct corollary is that subexponential DAD growth of $G(X)$ implies amenability, matching the known implication from subexponential asymptotic dimension growth to property A. The paper then proves that any $\\sigma$-compact étale groupoid with compact unit space and DAD growth at most $x^\\alpha$ for some $0<\\alpha<1$ is amenable; the proof constructs, from the growth bound, a generalized partition of unity whose $p$-th powers sum to one and whose variation along short arrows is uniformly small, and this yields an almost invariant measure.","pith_inferences":["If the apparent missing factor in inequality (5.1) is a typographical slip, the amenability theorem likely extends beyond $x^\\alpha$ to any subexponential growth function by refining the partition-of-unity estimates; if it is not, there may exist non-amenable groupoids with subexponential DAD growth, which would be a surprising separation.","The equivalence theorem suggests an untapped two-way bridge: dimension-growth bounds proved for group actions could be transplanted to coarse spaces, and coarse-geometric constructions could produce actions with prescribed growth.","The partition-of-unity proof has the shape of a quantitative Reiter condition, so the theorem may yield explicit estimates on the almost invariant measures in terms of the growth function, not just qualitative amenability."],"forward_implications":["For every bounded-geometry metric space, asymptotic dimension growth and the dynamic asymptotic dimension growth of the associated coarse groupoid have the same growth type, so results can flow across the coarse-groupoid dictionary.","Subexponential dynamic asymptotic dimension growth of a coarse groupoid forces the groupoid to be amenable, yielding new amenable groupoids from spaces such as geodesic coarse median spaces of finite rank and at most exponential volume growth.","Every $\\sigma$-compact étale groupoid with compact unit space and dynamic asymptotic dimension growth at most $x^\\alpha$ ($0<\\alpha<1$) is amenable, a strictly quantitative generalization of the finite-dimension amenability theorem that removes the freeness hypothesis.","A countable discrete group with subexponential asymptotic dimension growth admits a free, minimal action on the Cantor set whose dynamic asymptotic dimension growth is at most the same function, so examples like the wreath product $\\mathbb{Z}\\wr\\mathbb{Z}$ give actions with polynomial dynamic asymptotic dimension growth."],"supporting_citations":[{"why":"Introduced dynamic asymptotic dimension, supplying Lemma 4.1 and Proposition 7.1 that the growth version adapts.","marker":"[GWY17]"},{"why":"Built the coarse groupoid G(X) and proved property A of X is equivalent to amenability of G(X), used in Corollary 4.19.","marker":"[STY02]"},{"why":"Proved subexponential asymptotic dimension growth implies property A, connecting slow growth to amenability.","marker":"[Oza12]"},{"why":"Introduced asymptotic dimension growth and the comparison framework for growth functions, including wreath-product examples.","marker":"[Dra06]"},{"why":"Provided the equivalent characterizations of asymptotic dimension growth (their Lemma 2.6) underlying Theorem 4.16.","marker":"[ANWZ18]"},{"why":"Supplied the partition-of-unity method and the finite dynamical complexity amenability argument that Proposition 5.1 and Theorem 6.4 use.","marker":"[GWY24]"},{"why":"Provided coarse continuous length functions on étale groupoids, used to define DAD growth and to control compact subsets in Theorem 6.4.","marker":"[MW20]"},{"why":"Supplied the C*-algebraic construction of free, minimal Cantor actions used to prove Theorem 4.22.","marker":"[RS12]"}],"fun_headline_variants":["Subexponential dimension growth yields amenability","Slow dimension growth forces amenability in groupoids","Dimension growth equivalence for coarse groupoids","Amenability from subexponential DAD growth","Coarse groupoids: dimension growth meets amenability"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that a function that grows no faster than x to the alpha power (with alpha less than 1) still grows slowly enough that an error term involving the function divided by a large auxiliary scale can be made arbitrarily small; the paper only checks this by a sketch for one representative function, not for every admissible function.","fun_headline_variants_meta":{"raw":{"variants":["Subexponential dimension growth yields amenability","Slow dimension growth forces amenability in groupoids","Dimension growth equivalence for coarse groupoids","Amenability from subexponential DAD growth","Coarse groupoids: dimension growth meets amenability"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00043,"raw_usage":{"total_tokens":2186,"prompt_tokens":922,"completion_tokens":1264,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":538,"completion_tokens_details":{"reasoning_tokens":1195}},"tokens_in":538,"tokens_out":1264,"duration_ms":9133,"temperature":1.0,"reasoning_tokens":1195,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T05:54:42.416031+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the displayed inequality (5.1) in Proposition 5.1: as written the left-hand side lacks division by the auxiliary scale $N(R)$. With the natural correction, test whether the constant $c$ asserted to exist in Remark 5.7 actually makes the corrected inequality hold for the model function $f(x)=\\lceil x^{1/2}\\rceil$ at $R=1$ and $\\varepsilon=0.1$; if no such $c$ exists, or if some admissible $f\\le x^\\alpha$ violates the corrected estimate, the proof of Theorem 6.4 does not go through.","supporting_citations":[],"review_version":1}