{"id":"20b3ab8b-b94c-44dc-a633-036492a8e496","arxiv_id":"2510.22484","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For amenable lcσ group actions on compact metric spaces, diam-mean equicontinuity holds iff the maximal equicontinuous factor map is regular.","lead":"This paper proves that an action is diam-mean equicontinuous exactly when its projection to the maximal equicontinuous factor is regular (one-to-one up to measure zero), for amenable locally compact σ-compact group actions and without minimality. It also constructs a canonical maximal diam-mean equicontinuous factor, answering an open question.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"The paper's central claim is Theorem 1.1. I examined both directions. (⇒) Prop 7.9 uses Lemma 7.8, which bounds fibre mean diameter by sup_x Diam(B_δ(x)) for a Banach proximal map; the proof via finite subsets and translation is correct. (⇐) Theorem 7.11 lifts F-diam-mean equicontinuity along a regular map using Prop 7.10; the density argument there is sound, and Prop 6.9 upgrades 'all left Følner sequences' to full diam-mean equicontinuity. The weakest external input is the measure-lifting assertion in §2.6, needed for Remark 5.1 and for Prop 7.10's use of ν(Y0)=1. I verified the lifting is a standard Følner-averaging argument and holds for amenable lcσ groups; hence it is not a gap. The reader's verdict ACCEPT seems justified. Minor issues: (1) in Prop 2.5(ii) the displayed ε|K| should presumably be ε|F_n| for the subsequent inequality; (2) Prop 7.10 states 'left (or right)' but F-generic is only defined for left Følner sequences, and Theorem 7.11 only uses the left case; (3) Remark 6.6 explicitly leaves open whether one Følner sequence suffices for F-diam-mean equicontinuity to imply diam-mean equicontinuity, but the proof of Theorem 1.1 avoids this by establishing it for all left Følner sequences. None of these affect the central result.","tokens_in":23051,"tokens_out":27187,"duration_ms":251279,"concrete_test":"Write out the measure-lifting proof for a factor map π:X→Y between actions of an amenable lcσ group: for ergodic ν∈M_G(Y), take any μ0 with π_*μ0=ν, set μ_n = (1/|F_n|)∫_{F_n} g_*μ0 dg along a left Følner sequence, and verify that every weak* cluster point is invariant and satisfies π_*μ=ν. If this cannot be completed, Remark 5.1 and hence the equivalence (i)⇔(ii') in Theorem 5.2 would need a different justification; if it can, the central theorem is supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After a full pass over the proof of Theorem 1.1, I do not find a load-bearing gap. The least-secure step is the transfer in Remark 5.1: regularity of π is converted into ν(Y0)=1 for all ν∈M_G(Y), which needs the lifting of any invariant measure on Y to an invariant measure on X. This is asserted in §2.6 to follow from amenability. The lifting is correct: π_*:M(X)→M(Y) is surjective (its image is compact convex and contains all Dirac masses), so for ν choose μ0 with π_*μ0=ν; for a Følner sequence F_n, cluster points of (1/|F_n|)∫_{F_n} g_*μ0 dg are invariant and project to ν. Thus the step is justified inside the paper's amenable-lcσ setting and does not threaten Theorem 1.1. The other cited inputs, FGL22's Prop 7.3/7.4, are standard for these groups. Minor issues (e.g., a small typo in the ε factor in Prop 2.5(ii), and the unused 'right Følner' variant in Prop 7.10 where F-generic is only defined for left Følner) do not affect the main result.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the mean diameter of subsets in actions of amenable locally compact sigma-compact groups on compact metric spaces. It defines a Følner-sequence-dependent mean diameter Diam_F and a uniform mean diameter Diam obtained by taking the supremum over all left Følner sequences. The central result, Theorem 5.2, characterizes regular factor maps as exactly those for which every fibre has mean diameter zero, with equivalent formulations in terms of upper Banach density and in terms of a single Følner sequence. Building on this, Theorem 1.1 proves that an action is diam-mean equicontinuous if and only if the factor map onto its maximal equicontinuous factor is regular, with no minimality assumption. The paper then establishes stability of regular factor maps under products and composition, gives a counterexample showing that hyperspace maps do not preserve regularity, and constructs a maximal diam-mean equicontinuous factor.","tokens_in":23367,"tokens_out":13004,"duration_ms":121320,"significance":"If correct, Theorem 1.1 resolves an open question in the area, extending earlier results for minimal actions of Z, abelian lc sigma groups, and countable amenable groups to all actions of amenable lc sigma groups. The proof strategy is genuinely different from earlier approaches, avoiding frequent stability and working directly with mean diameters. The characterization of regular factor maps by vanishing mean diameter of fibres is a clean and useful tool. The paper is careful about the non-minimal setting, and the measure-lifting step, which is the least obvious point, is justified by amenability of the acting group. I found no circularity: the main theorem depends on external results such as [FGL22] and [Lin01], and the self-citations are auxiliary. The proofs contain explicit density estimates, and the construction of the maximal diam-mean equicontinuous factor is self-contained. Overall, this is a substantial and convincing contribution.","major_comments":[],"minor_comments":[{"comment":"The displayed estimate \"|F_n Δ K^{-1}F_n| ≤ ε|K|\" is followed by \"|K^{-1}F_n| ≤ (1+ε)|F_n|\". As written, this does not follow directly from the previous display; it holds only for sufficiently large n, using that |F_n| → ∞. The argument is correct, but the inequality should be stated with an explicit \"for large n\" or with ε|F_n| in place of ε|K|.","section":"§2.7, proof of Prop. 2.5(ii)"},{"comment":"The proposition is stated for a \"left (or right) Følner sequence\", but F-genericity is defined only for left Følner sequences in §2.6. If right Følner sequences are intended, the definition of F-generic should be extended accordingly, or the parenthetical should be removed.","section":"§7.3, Prop. 7.10"},{"comment":"The sentence \"From Proposition 7.4 we observe that π(x) is F-generic\" should explicitly cite Proposition 7.4(ii), since part (i) concerns points in the maximal support. The application is correct because Y is mean equicontinuous, but the citation is currently ambiguous.","section":"§7.3, proof of Thm. 7.11(i)"},{"comment":"The assertion that for every ergodic ν on Y there exists an invariant μ on X with π_*μ = ν is stated as a \"straightforward argument\". Given that this step is load-bearing in Remark 5.1, a short proof or a precise reference would improve the exposition.","section":"§2.6"},{"comment":"The inequality 0 < d_H(A,B) ≤ Diam(H(π)^{-1}(Y)) is true, but a one-line justification would be helpful: since A and B are fixed by the action, every term in the Følner average is at least d_H(A,B).","section":"§8.3, Example 8.3"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main thing to know: this paper proves what was explicitly open — for actions of amenable lcσ groups on compact metric spaces, diam-mean equicontinuity is equivalent to regularity of the factor map onto the maximal equicontinuous factor, with no minimality assumption. It also characterizes regular factor maps as those with zero mean diameter in every fibre (Theorem 5.2) and builds a maximal diam-mean equicontinuous factor. That is a genuine advance over the earlier minimal-Z, Abelian-lcσ, and countable-amenable-with-local-Bronstein results. I read the proof of the core equivalence and it holds up: Proposition 7.9 plus Theorem 7.11, with the mean-diameter machinery in Sections 4–5 doing the heavy lifting. The measure-lifting step in Remark 5.1, which the reader flagged as the weakest point, is in fact justified inside the paper by a standard Krylov–Bogolyubov argument in §2.6; the stress-test note confirms it. So that concern does not land. The paper is well grounded in FGL22 and Lindenstrauss's ergodic theorem, and the few self-citations ([Hau24], [FGH25], [CH26]) are auxiliary, not load-bearing. The 'regular iff diam-mean proximal iff F-diam-mean proximal' theorem is a clean and useful formulation, and the proof via the hyperspace action is elegant. Soft spots are minor: dense notation, a pronoun slip in §7.3, an overly brief unique-ergodicity assertion in Example 8.3, and a small typo in Proposition 2.5(ii)'s epsilon factor. None of these affect the central argument. The one thing I would press on in referee report is the Følner-sequence dependence: Remark 6.6 notes that F-diam-mean equicontinuity for a single sequence does not imply diam-mean equicontinuity, and Proposition 6.9's equivalence uses 'for all sequences'. That is handled correctly, but a reader could easily miss it, so a bit more signposting would help. Who is this for? Anyone working on mean equicontinuity, regular extensions, or non-minimal topological dynamics. It deserves a serious referee — the result is important enough for the subfield, and the proofs are explicit enough to check. My verdict: accept after minor revision. I would cite it.","headline":"Solid, honest paper that settles the non-minimal diam-mean equicontinuity characterization for amenable lcσ groups and adds a clean regularity criterion; minor presentation issues only.","tokens_in":23807,"tokens_out":721,"would_cite":true,"duration_ms":8794,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37B05","37B25","37A05"],"pacs":[],"model":"deepseek-v4-flash","headline":"An action of an amenable locally compact sigma-compact group is diam-mean equicontinuous exactly when the map onto its maximal equicontinuous factor is regular.","keywords":["diam-mean equicontinuity","regular factor map","mean diameter","amenable group","maximal equicontinuous factor","diam-mean proximality","Følner sequence","non-minimal actions"],"falsifier":"Take the shift action of the free group on two generators on the Cantor set, which is non-amenable, and compute the mean diameter of the fibres over the maximal equicontinuous factor. If the fibres all have mean diameter zero while some point has no neighbourhood of arbitrarily small mean diameter, the equivalence fails outside the amenable setting. Within the paper's setting, a counterexample would be an amenable-lcσ action whose maximal-equicontinuous-factor map is regular but some ball has mean diameter bounded below.","tokens_in":1249,"feed_emoji":"📏","tokens_out":1779,"duration_ms":63280,"temperature":0.7,"pith_summary":"The paper establishes that diam-mean equicontinuity, a quantitative average version of equicontinuity, is not a separate dynamical phenomenon: for actions of amenable locally compact sigma-compact groups on compact metric spaces, it is equivalent to the factor map onto the maximal equicontinuous factor being regular, meaning its injectivity points carry full measure for every invariant measure. This answers an open question for non-minimal actions and removes minimality assumptions from earlier results. The bridge is a new characterization of regular factor maps as diam-mean proximal maps: every fibre has mean diameter zero. As a consequence, the paper constructs a maximal diam-mean equicontinuous factor for every such action and shows the class is closed under countable products.","feed_headline":"Diam-mean equicontinuity equals regular fibres","feed_subtitle":"New theorem drops minimality and applies to all amenable locally compact sigma-compact groups.","key_machinery":"The mean diameter Diam(A) = sup_F limsup_n (1/|F_n|) ∫_{F_n} diam(g.A) dg, taken over all left Følner sequences of the acting group, is the operative quantity. It is shown to equal an infimum over compact sets of a supremum over shifts, and it turns the measure-theoretic notion of regularity into a geometric, Følner-average condition. The proof rests on Theorem 5.2, which equates regularity of a factor map with diam-mean proximality — zero mean diameter of every fibre — together with the ability to lift invariant measures from the quotient, a property provided by amenability.","core_discovery":"The central claim is Theorem 1.1: an action of an amenable lcσ group on a compact metric space is diam-mean equicontinuous if and only if the factor map π_eq: X → X_eq onto its maximal equicontinuous factor is regular. The paper proves this via Theorem 5.2, which states that a factor map π: X → Y is regular if and only if every fibre π^{-1}(y) has mean diameter zero, equivalently the upper Banach density of times g with diam(g.π^{-1}(y)) > ε is at most ε. This characterization, together with a measure-lifting argument supplied by amenability, yields both directions: diam-mean equicontinuous actions have regular maximal equicontinuous factor, and regular extensions of equicontinuous actions a","pith_inferences":["The equivalence suggests diam-mean equicontinuity is best understood as 'regular extension of an equicontinuous action'; in aperiodic-order settings where actions of R^n appear, this yields a measure-one injectivity criterion that needs no minimality.","The proof's reliance on measure lifting indicates the theorem may be sharp: for non-amenable acting groups, regularity could cease to imply the quotient-side measure condition, so the characterization likely requires a modified notion of regularity outside the amenable class.","The paper leaves open whether F-diam-mean equicontinuity for a single Følner sequence already implies diam-mean equicontinuity in this generality; a positive answer would let the whole theory be checked from one averaging sequence."],"forward_implications":["Diam-mean equicontinuity can be verified by inspecting only the maximal equicontinuous factor: one checks whether the fibre diameters shrink to zero in mean.","The characterization covers non-minimal actions and actions of general amenable locally compact sigma-compact groups, not just minimal Z-actions or abelian groups.","Regular factor maps are preserved under countable products and under composition and decomposition, so diam-mean equicontinuity is stable under countable products.","Every action of this class has a unique, up to conjugacy, maximal diam-mean equicontinuous factor, and its maximal equicontinuous factor is the maximal equicontinuous factor of that factor.","Regular Toeplitz-type extensions of odometers become concrete examples of diam-mean equicontinuous actions."],"fun_headline_variants":["Mean diameter zero characterizes regular factors","Regular fibres decide diam-mean equicontinuity","Diam-mean equicontinuity iff regular maximal factor","Mean diameter and regularity unify equicontinuity","Theorem links fibre regularity to equicontinuity"],"cache_read_input_tokens":25216,"weakest_assumption_plain":"The load-bearing premise is that every ergodic invariant measure on the quotient lifts to an invariant measure on the extension; this follows from amenability and is exactly what lets regularity of the factor map be detected from the quotient side, and without it the central equivalence can fail.","fun_headline_variants_meta":{"raw":{"variants":["Mean diameter zero characterizes regular factors","Regular fibres decide diam-mean equicontinuity","Diam-mean equicontinuity iff regular maximal factor","Mean diameter and regularity unify equicontinuity","Theorem links fibre regularity to equicontinuity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001154,"raw_usage":{"total_tokens":4563,"prompt_tokens":634,"completion_tokens":3929,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":378,"completion_tokens_details":{"reasoning_tokens":3852}},"tokens_in":378,"tokens_out":3929,"duration_ms":26594,"temperature":1.0,"reasoning_tokens":3852,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T08:04:38.149290+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the shift action of the free group on two generators on the Cantor set, which is non-amenable, and compute the mean diameter of the fibres over the maximal equicontinuous factor. If the fibres all have mean diameter zero while some point has no neighbourhood of arbitrarily small mean diameter, the equivalence fails outside the amenable setting. Within the paper's setting, a counterexample would be an amenable-lcσ action whose maximal-equicontinuous-factor map is regular but some ball has mean diameter bounded below.","supporting_citations":[],"review_version":1}