{"id":"cd099d14-71d4-449e-ac08-40450369e4f4","arxiv_id":"2501.07038","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For minimal abelian group actions, frequent (m+1)-stability is equivalent to almost m-to-one maximal equicontinuous factors, and diam-mean (m+1)-equicontinuity to almost surely m-to-one factors.","lead":"This paper introduces multi-point versions of two notions that measure how close a dynamical system is to being highly ordered, and proves they exactly describe systems whose simplest ordered quotient sees each point only a bounded number of times. It generalizes earlier one-point characterizations to arbitrary finite fiber sizes and to general abelian group actions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 5.4's key density estimate is not implied by (7): the proof passes from F-density for base point y* to lower density along a_nF_n for base point β(a_n,y*), which requires an unstated uniform/unique-ergodicity argument; Theorem 5.6's central direction is therefore incomplete as written…","rationale":"The reader's weakest_assumption identifies the same location: the transition from (7) to the shifted-Følner density estimate in Proposition 5.4. I agree that this is the most load-bearing gap in the paper. However, the concern is about completeness of the proof rather than falsity of the theorem: the MEF is uniquely ergodic, and the bad set A is open, so a uniform lower Banach density estimate can likely be obtained by the Uniform Ergodic Theorem and standard approximation arguments. Thus the correct verdict remains CONDITIONAL, as the reader already concluded; no adjustment to the verdict is needed. The main theoretical claims are plausible and the flaws are localized and repairable.","tokens_in":22314,"tokens_out":17189,"duration_ms":187382,"concrete_test":"Analytically re-derive the density estimate in Proposition 5.4 using Theorem 1.14 instead of (7): prove that for the open set A = {π*diam_{m+1} > 5ε} in the uniquely ergodic MEF, one has BDF({g : β(g,z) ∈ A}) ≥ ν(A) for every z and every Følner sequence F, by approximating 1_A with continuous functions on compact subsets. If this uniform lower Banach density estimate holds, insert it into Proposition 5.4 and re-check the contradiction step; that closes the gap. If a counterexample is found (for instance, an open A with positive-measure boundary in an irrational rotation), then Theorem 5.6 needs a different argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The engine of Theorem 5.6 is Proposition 5.4, whose proof contains the line: after defining H = {g | π*diam_{m+1}(β(g,y*)) > 5ε}, it asserts that (7) implies lim inf_n m(a_nF_n ∩ H)/m(a_nF_n) > 2ε. This is not a consequence of (7) as stated. Two gaps are present: (a) lower F-density is not generally preserved under arbitrary shifts a_nF_n, even for sets of positive F-density; (b) more importantly, a_nF_n ∩ H is not a fixed translate of the set measured in (7), because membership uses the orbit of β(a_n,y*) rather than y*. The natural repair is to use the unique ergodicity of the MEF and the fact that A = {π*diam_{m+1} > 5ε} is open: for any compact K ⊂ A one can choose continuous f with 1_K ≤ f ≤ 1_A and apply the Uniform Ergodic Theorem 1.14, yielding a lower Banach density bound uniformly in the base point. The text cites only the pointwise Lindenstrauss theorem and does not supply this argument. Hence the proof of Proposition 5.4 has a missing step that is load-bearing for Theorem 5.6, although the statement is likely correct and repairable.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces multivariate versions of frequent stability and diam-mean equicontinuity for minimal topological dynamical systems with a sigma-compact locally compact abelian acting group, using the multivariate distance D_m and the associated diameter diamm. The main results are Theorem 4.5, which asserts that frequent (m+1)-stability is equivalent to the factor map to the maximal equicontinuous factor being almost m:1, and Theorem 5.6, which asserts that diam-mean (m+1)-equicontinuity is equivalent to that factor map being almost surely m:1. The proofs combine hyperspace techniques, ergodic decomposition, and the structure of the MEF, and they generalize earlier univariate results of Xu-Hu and García-Ramos-Jäger-Ye.","tokens_in":22623,"tokens_out":21081,"duration_ms":199908,"significance":"If the results are correct, they provide a coherent multivariate hierarchy in which finite-to-one behavior of the MEF extension is characterized by weak rigidity properties of the system. The extension to general sigma-compact LCA acting groups and the use of Følner and Banach densities are substantial, and the paper is largely self-contained. Theorem 4.5 is supported by a coherent proof, and the main difficulty is localized in Proposition 5.4, where a density estimate needs a uniform-ergodicity argument. The statements are likely correct and repairable, so the result would be a useful contribution to the study of weak equicontinuity and MEF extensions.","major_comments":[{"comment":"The step after Eq. (7) is not justified. From D_F(H)>2ε for H={g : π*diam_{m+1}(β(g,y*))>5ε} one cannot conclude that lim inf_n m(a_nF_n∩H)/m(a_nF_n)>2ε for the arbitrary shifts a_n chosen later in the proof; F-density is not invariant under left translations of general Følner sets, and the proof supplies no uniform-in-base-point estimate. Moreover, Theorem 1.18 requires a tempered Følner sequence, and none is selected. This matters because Corollary 5.5 and the (iii)⇒(i) direction of Theorem 5.6 both rely on Proposition 5.4. The gap is repairable: since the MEF is uniquely ergodic and A={y : π*diam_{m+1}(y)>5ε} has ν(A)>2ε, one can choose a compact K⊂A and a continuous f with 1_K≤f≤1_A and apply the Uniform Ergodic Theorem 1.14 and Corollary 1.15 to obtain a uniform lower density bound for H valid along every translated Følner set a_nF_n.","section":"Proposition 5.4"},{"comment":"As printed, the 'Banach' and 'weakly' clauses are identical: both require BDF(T^{(m)}_{δ,ε}(x))<1 in Definition 4.1 and both require BDF(T^{(m)}_{δ,ε}(x))<ε in Definition 5.1. This cannot be the intended distinction, and it is load-bearing because Theorem 4.5(iii) and Theorem 5.6(iii) assert an existential weak version that must be weaker than the universal Banach version. The notation in Definition 1.10 also uses the same symbol for the upper and lower F-Banach densities, making Lemma 1.11 and all later density inequalities ambiguous. The authors should restore the intended upper/lower density symbols and make the proofs of (iii)⇒(i) consistent with the corrected definitions.","section":"Definitions 4.1 and 5.1"},{"comment":"Even setting aside the density-translation gap, Proposition 5.4 yields only a limsup bound along one chosen sequence of shifts. Such a bound directly controls the upper F-Banach density of the bad set, not the lower F-Banach density. If the Banach condition in Definition 5.1 is intended to use the lower F-Banach density, Corollary 5.5 does not follow as written; if it is intended to use the upper F-Banach density, the notation in Definitions 1.10 and 5.1 must be made unambiguous. The central proof of Theorem 5.6 can be reorganized to avoid relying on this corollary, but the statement itself needs correction.","section":"Corollary 5.5"}],"minor_comments":[{"comment":"The symbols for upper and lower F-density and upper and lower F-Banach density are not visually distinguished in the typeset text, so Lemma 1.11 as displayed is not a meaningful inequality chain; please use consistently distinguished notations such as \\overline{BD}_F and \\underline{BD}_F.","section":"Section 1.1"},{"comment":"The text first states 'By the Lindenstrauss Ergodic Theorem 1.18, there is y*' and then says 'Pick any y*'; the second occurrence should be removed or reworded, since the later construction must use the y* supplied by the ergodic theorem.","section":"Proposition 5.4"},{"comment":"When applying the Lindenstrauss Ergodic Theorem to a tempered subsequence of FB*, the authors should explicitly say that the tempered subsequence is chosen so that the limsup bound from Proposition 5.4 is still realized along that subsequence; otherwise the contradiction M_e^*(Ω^c)>2ε is not directly tied to the density estimate.","section":"Theorem 5.6, proof of (iii)⇒(i)"},{"comment":"In the proof of the lower bound, the inductive construction gives D_m(a_1,...,a_m)≥ε rather than >ε, since the selected points avoid open ε-balls; the conclusion diamm(A)≥Em(A) still follows after taking the supremum over ε<Em(A), but the strict inequality should be corrected.","section":"Lemma 1.21"}],"recommendation":"major_revision","confidential_remarks":"The main gap is localized and repairable, and the central claims are likely correct, so I recommend major revision rather than rejection. The definition/notation ambiguity in 'weak' versus 'Banach' conditions should be resolved before acceptance, as it affects the precise statements of Theorems 4.5 and 5.6."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version. The paper does something real: it defines multivariate frequent m-stability and diam-mean m-equicontinuity and proves that, for minimal actions of sigma-compact LCA groups, the first is equivalent to almost m:1 extension of the MEF and the second to almost surely m:1. Theorem 4.5 looks correct. The proof is a clean generalization of Xu-Hu, and the new characterization is a genuine advance over the m=1 case. The companion paper supplies the multivariate distance and polygon inequality, which are proved there, so the self-citation is not a problem.\n\nThe weak spot is Proposition 5.4, which carries Theorem 5.6. The proof defines H from a point y* and asserts that (7) gives lim inf of m(a_nF_n ∩ H)/m(a_nF_n) > 2ε. That doesn't follow. First, lower F-density is not preserved under arbitrary shifts of the Følner sequence. Second, the shifted average is really measuring the return set for β(a_n,y*), not y*, because the base point changes when you move from F_n to a_nF_n. The likely repair is to use unique ergodicity of the MEF and the Uniform Ergodic Theorem, as the stress-test note says, to get a lower Banach density bound uniformly in the base point. That missing argument is load-bearing, so Theorem 5.6 is incomplete as written. I still think the statement is true and repairable.\n\nMinor issues: the 'weak' and 'Banach' definitions are textually identical in both sections, which makes the weak-versus-Banach distinction in the theorem statements meaningless as written. Probably the weak version should use the un-shifted upper density. Also, in Proposition 5.4 the phrase 'Pick any y*' appears twice, right after y* was chosen by the ergodic theorem; likely a typo.\n\nBottom line: this is a serious paper, not a desk reject. Theorem 4.5 alone justifies refereeing. A referee should ask for a repaired proof of Proposition 5.4 and a cleanup of the weak/Banach definitions. I'd engage with it.","headline":"Real multivariate extension with a solid first theorem; the second theorem relies on a missing density-translation argument in Proposition 5.4.","tokens_in":23102,"tokens_out":2138,"would_cite":true,"duration_ms":20488,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37B05","37A25","37A30"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that multivariate frequent stability and diam-mean equicontinuity characterize finite-to-one extensions of the maximal equicontinuous factor in minimal systems.","keywords":["multivariate frequent stability","diam-mean equicontinuity","maximal equicontinuous factor","finite-to-one extensions","Følner sequences","Banach density","minimal topological dynamics","regionally proximal relation"],"falsifier":"Take a minimal system and Følner sequence for which the MEF map is not almost surely $m:1$, and compute the lower density inside the shifted windows $a_nF_n$ of the return-time set $H=\\{g:\\pi^*\\mathrm{diam}_{m+1}(\\beta(g,y^*))>5\\varepsilon\\}$ used in Proposition 5.4; a concrete system with value $\\le 2\\varepsilon$ would disprove the proof's key estimate, while a case with the expected lower bound would support the argument as written.","tokens_in":22079,"feed_emoji":"🌀","tokens_out":13388,"duration_ms":107070,"temperature":0.7,"pith_summary":"This paper introduces multivariate versions of two weak rigidity notions—frequent stability and diam-mean equicontinuity—and uses them to characterize exactly how many preimages the maximal equicontinuous factor (MEF) can have. The main results, proved for minimal systems with a $\\sigma$-compact, locally compact, abelian acting group, state that frequent $(m+1)$-stability is equivalent to the MEF factor map being almost $m$-to-one, while diam-mean $(m+1)$-equicontinuity is equivalent to the factor map being almost surely $m$-to-one. This matters because many natural systems, such as constant-length substitution subshifts, are finite-to-one extensions of their MEF rather than one-to-one extensions, and previous multivariate work covered equicontinuity and mean equicontinuity but not these two rigidity properties. If correct, the paper turns a fiber-counting question into a checkable dynamical condition on iterates of small balls, and vice versa.","feed_headline":"Multivariate stability now characterizes finite-to-one MEF fibers","feed_subtitle":"In minimal systems, (m+1)-stability means almost m:1; diam-mean means almost surely m:1.","key_machinery":"The central object is the multivariate diameter $\\mathrm{diam}_m$ built from the minimal-pairwise-distance function $D_m(x_1,\\ldots,x_m)=\\min_{i<j} d(x_i,x_j)$. It turns the statement 'all points of a small ball stay well separated' into a continuous, real-valued observable on the hyperspace $(\\mathcal{K}(X),d_H)$ of compact subsets, so that the induced action on the hyperspace can be studied through Følner averaging, Banach densities, and the pointwise ergodic theorem for tempered Følner sequences. The second load-bearing mechanism is the regional proximal relation $Q_m$: by Theorem 2.9, in the abelian minimal setting an $m$-tuple lies in $Q_m$ exactly when all its entries are in one MEF fiber, so fiber cardinality becomes a proximality statement. The dichotomies in Section 3 then force a uniform positive lower bound on the $(m+1)$-diameter of fibers exactly when the relevant finiteness condition fails.","core_discovery":"For a minimal topological dynamical system $(X,\\alpha,G)$ with $\\sigma$-compact locally compact abelian $G$, the paper claims that two multivariate rigidity properties are exactly the finite-to-one extension properties of the maximal equicontinuous factor (MEF). Frequent $(m+1)$-stability—small balls having, for a positive-density set of group elements, iterates whose $(m+1)$-diameter is small—is claimed equivalent to the factor map $\\pi\\colon X\\to Y$ being almost $m:1$, meaning the set of points with at most $m$ preimages is residual. Diam-mean $(m+1)$-equicontinuity, where the average of that diameter over every Følner sequence is small, is claimed equivalent to $\\pi$ being almost surely $m:1$, meaning the set of points with at most $m$ preimages has full Haar measure in the MEF. The paper proves both equivalences in full, Theorem 4.5 for frequent stability and Theorem 5.6 for diam-mean equicontinuity, with weak one-point versions implying the strong uniform versions.","pith_inferences":["A consequence the author leaves implicit is that for systems whose MEF extension is finite-to-one but not one-to-one, such as constant-length substitution subshifts, the theorem predicts which multivariate diameter condition holds solely from the fiber bound, giving a direct laboratory for the distinction between the residual and full-measure versions of $m:1$.","The proof's reliance on $Q_2=Q^*$ suggests that the equivalence may fail for non-abelian acting groups, where higher-order regionally proximal tuples need not be determined by pairwise proximality; extending the result would require a different bridge between fibers and multivariate diameter.","The paper's Banach-density formulation invites a possible sharpening: since the pointwise ergodic theorem only requires tempered Følner sequences, the quantifier 'for every Følner sequence' in the definitions might be replaceable by 'for every tempered Følner sequence' without changing the class of systems.","A testable extension is to check whether the two dichotomies—residual versus positive lower bound in Lemma 3.12, and full measure versus positive measure in Lemma 3.8—form a strict hierarchy in $m$ for symbolic systems with known MEF fiber structure, such as constant-length substitutions or model sets."],"forward_implications":["For minimal $\\mathbb{Z}$-actions, the univariate case $m=1$ returns the known characterizations: frequent stability is almost $1:1$ and diam-mean equicontinuity is almost surely $1:1$.","Weak one-point versions are enough: if a single point is weakly $(m+1)$-stable for a single Følner sequence, then every point is Banach $(m+1)$-stable for every Følner sequence; the same holds for diam-mean $(m+1)$-equicontinuity.","The dichotomy lemmas imply a clean gap: if almost $m:1$ fails, the $(m+1)$-diameter of every MEF fiber is bounded below by a positive constant, giving a quantitative obstruction to frequent stability.","The results hold for every $\\sigma$-compact locally compact abelian acting group, not just $\\mathbb{Z}$-actions, so the same finite-to-one characterization applies to actions of $\\mathbb{R}^d$, tori, and $p$-adic groups.","Constant-length substitution subshifts, whose MEF fibers are finite but not necessarily single points, become test cases for the new notions instead of falling outside the one-to-one framework."],"supporting_citations":[{"why":"It defines the multivariate distance $D_m$ and proves the polygon inequality that the multivariate diameter of this paper uses as its basic estimate.","marker":"[5]"},{"why":"It introduced the two univariate notions and proved the diam-mean almost-surely one-to-one characterization that Theorem 5.6 extends.","marker":"[14]"},{"why":"It proved that minimal frequent stability is equivalent to an almost one-to-one MEF extension, and its hyperspace argument is the template for the converse direction in Theorem 4.5.","marker":"[26]"},{"why":"It supplies the pointwise ergodic theorem for tempered Følner sequences, used in both converse directions to pass from orbit averages to ergodic measures on the hyperspace.","marker":"[21]"},{"why":"It provides the result that $Q_m$ tuples coincide with points in the same MEF fiber in the abelian minimal setting, the bridge from fiber cardinality to proximality.","marker":"[3]"},{"why":"It established the measure-theoretic isomorphism characterization of mean equicontinuity that motivates the almost-surely finite-to-one formulation and its dichotomy.","marker":"[8]"},{"why":"It supplies constant-length substitution examples whose MEF extensions are finite-to-one, the concrete systems the new notions are designed to capture.","marker":"[6]"}],"fun_headline_variants":["Frequent and diam-mean stability: finite fibers or full measure","Finite-to-one MEF extensions: new stability characterizations","Stability in minimal systems: almost m-to-one MEF maps","New equivalence: multivariate stability and MEF fiber size","MEF fibers: frequent stability vs diam-mean equicontinuity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the diam-mean characterization assumes, without stating or proving it, that a set of group elements with positive lower F-density still has positive lower density in every shifted Følner set $a_nF_n$, a shift-invariance that need not hold for arbitrary Følner sequences.","fun_headline_variants_meta":{"raw":{"variants":["Frequent and diam-mean stability: finite fibers or full measure","Finite-to-one MEF extensions: new stability characterizations","Stability in minimal systems: almost m-to-one MEF maps","New equivalence: multivariate stability and MEF fiber size","MEF fibers: frequent stability vs diam-mean equicontinuity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000404,"raw_usage":{"total_tokens":2113,"prompt_tokens":962,"completion_tokens":1151,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":578,"completion_tokens_details":{"reasoning_tokens":1062}},"tokens_in":578,"tokens_out":1151,"duration_ms":64839,"temperature":1.0,"reasoning_tokens":1062,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:51:33.992437+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a minimal system and Følner sequence for which the MEF map is not almost surely $m:1$, and compute the lower density inside the shifted windows $a_nF_n$ of the return-time set $H=\\{g:\\pi^*\\mathrm{diam}_{m+1}(\\beta(g,y^*))>5\\varepsilon\\}$ used in Proposition 5.4; a concrete system with value $\\le 2\\varepsilon$ would disprove the proof's key estimate, while a case with the expected lower bound would support the argument as written.","supporting_citations":[{"cited_title":"Multivar iate mean equicontinuity for finite-to-one topomorphic extensions, 2024","cited_arxiv_id":null,"evidence_quote":"It defines the multivariate distance $D_m$ and proves the polygon inequality that the multivariate diameter of this paper uses as its basic estimate."},{"cited_title":"Mean equicontinuity, almost automor- phy and regularity","cited_arxiv_id":null,"evidence_quote":"It introduced the two univariate notions and proved the diam-mean almost-surely one-to-one characterization that Theorem 5.6 extends."},{"cited_title":"Minimal frequently stable is almost auto morphic, 2024","cited_arxiv_id":null,"evidence_quote":"It proved that minimal frequent stability is equivalent to an almost one-to-one MEF extension, and its hyperspace argument is the template for the converse direction in Theorem 4.5."},{"cited_title":"Pointwise theorems for amenable groups","cited_arxiv_id":null,"evidence_quote":"It supplies the pointwise ergodic theorem for tempered Følner sequences, used in both converse directions to pass from orbit averages to ergodic measures on the hyperspace."},{"cited_title":"A group theoretic condition in topological dyn amics","cited_arxiv_id":null,"evidence_quote":"It provides the result that $Q_m$ tuples coincide with points in the same MEF fiber in the abelian minimal setting, the bridge from fiber cardinality to proximality."},{"cited_title":"Isomorphic extensions and applications","cited_arxiv_id":null,"evidence_quote":"It established the measure-theoretic isomorphism characterization of mean equicontinuity that motivates the almost-surely finite-to-one formulation and its dichotomy."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies constant-length substitution examples whose MEF extensions are finite-to-one, the concrete systems the new notions are designed to capture."}],"review_version":1}