{"id":"55db03c5-7a0b-4d89-aae7-eec6c6eb80b4","arxiv_id":"1908.05207","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For minimal systems, frequent stability is equivalent to the maximal equicontinuous factor being almost one-to-one, and diam-mean equicontinuity is equivalent to that factor being regular.","lead":"The authors prove that two natural forms of stability in rigid dynamical systems match two missing levels in the hierarchy built around group rotations. The result gives mathematicians a complete ladder from perfect rotations to much more complicated strictly ergodic models.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 4.9's uniform-ergodicity step for 1_{Gε} is not justified and is load-bearing for Theorem 4.12; the boundary of Gε is never shown to be Haar-null.","rationale":"The reader's weakest assumption concerns the unproved transitive characterization after Definition 4.6, which is used in the positive-entropy example of Section 5. That is a real gap, but it is secondary to the paper's main theorem. The most load-bearing issue for the central claim is instead the uniform-ergodicity step in Proposition 4.9: it is the only passage that proves one direction of Theorem 4.12, and it invokes a principle that needs a null-boundary hypothesis that is never stated or verified. I did not find a reason to believe Theorem 4.12 itself is false; the gap appears patchable by selecting the finite cover radii so that the sphere boundaries have zero Haar measure. Because the paper's main equivalence is therefore not fully established by the written proof, conditional acceptance remains the right verdict. My concern is distinct from the reader's singled-out assumption, so I disagree with the reader's identification of the weakest point, but the overall verdict stays unchanged.","tokens_in":16006,"tokens_out":33480,"duration_ms":381497,"concrete_test":"Test the uniform-ergodicity assertion in Proposition 4.9 on a concrete minimal rotation, e.g., T(x)=x+α mod 1 on the circle, with an open set G whose boundary has positive Lebesgue measure (for instance, an open set whose complement is a fat Cantor set). Compute sup over starting points y and window lengths K of (1/K)∑_{i=0}^{K−1} 1_G(T^i y). If this quantity does not converge to ν(G) as K→∞, the asserted uniform lower bound is false for the class of sets used in the proof. A simpler settlement: check whether the finite cover Gε in Proposition 4.9 can always be chosen with ν_eq(∂Gε)=0; if the proof cannot be amended to ensure this, the uniform-ergodicity step remains unjustified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 4.12, the central claim, is built on Proposition 4.9, which establishes regular → Banach diam-mean equicontinuous. The proof fixes an open set Gε with ν_eq(Gε) > 1−ε and then asserts: 'By uniform ergodicity, there exists L > 0 such that (1/K)∑_{i=j}^{j+K−1} 1_{Gε}(T_eq^i y) ≥ 1−ε for all j ∈ N, K ≥ L and y ∈ X_eq.' This does not follow from unique ergodicity as written. Uniform ergodicity gives uniform convergence of ergodic averages for continuous functions, and more generally for Riemann-integrable functions whose discontinuity set has measure zero. The indicator 1_{Gε} is discontinuous on ∂Gε, and the proof never verifies that ν_eq(∂Gε) = 0. For finite unions of metric balls in a compact abelian group, sphere boundaries need not be Haar-null in general, and without this the uniform lower bound on visits to Gε is not justified. If that bound fails for some long windows, the subsequent conclusion that all sufficiently long iterates of a small ball have small diameter collapses, so the direction (3) ⇒ (2) of the central theorem is not established by the given argument. This is a proof gap rather than a demonstrated counterexample, and it is likely repairable by choosing the finite cover radii so that all sphere boundaries are null; but as written the proof is incomplete.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies minimal topological dynamical systems with discrete spectrum, classified according to the properties of the maximal equicontinuous factor map π_eq. The main results are: (1) a characterization of almost automorphic minimal mean equicontinuous systems via 'frequent stability' (Theorem 3.6); (2) the central Theorem 4.12, which shows that for minimal systems, diam-mean equicontinuity, Banach diam-mean equicontinuity, and regularity of π_eq are equivalent; (3) a construction of a transitive almost diam-mean equicontinuous system with positive topological entropy (Theorem 5.6 and Corollary 5.7); and (4) a positive answer to Furstenberg's multiple recurrence question for mean equicontinuous systems (Theorem 6.1). The proofs use standard tools: Baire category, unique ergodicity, the Birkhoff ergodic theorem, measurable selection, and the pointwise multiple ergodic theorem.","tokens_in":16308,"tokens_out":18191,"duration_ms":170012,"significance":"If correct, the results complete a natural hierarchy for strictly ergodic discrete-spectrum systems, placing diam-mean equicontinuity exactly at the level where π_eq is regular. This sharpens previous work of Li–Tu–Ye, Downarowicz–Glasner, and García-Ramos, and the positive-entropy transitive example is a notable new phenomenon. The Furstenberg question result for mean equicontinuous systems is also a valuable contribution. The paper is generally well structured and builds on established references. However, two load-bearing proof points need attention: an unproved characterization in Section 4.1 and a false measure assertion in Proposition 4.11. Both are repairable, so the central claims are likely sound.","major_comments":[{"comment":"The statement 'A transitive t.d.s. is almost diam-mean equicontinuous if and only if there exists a diam-mean equicontinuity point' is asserted without proof. The 'if' direction is used in Theorem 5.6 to conclude that the constructed system is almost diam-mean equicontinuous from the fact that x is such a point. This is load-bearing for Corollary 5.7, and the equivalence is not obvious. Please provide a proof or a precise reference, in particular explaining why the existence of one such point implies the set of such points is residual.","section":"Section 4.1 (after Definition 4.6)"},{"comment":"The proof states 'Assume that π_eq is not regular. Then ν_eq({y : diam(π^{-1}_eq(y)) > 0}) = 1.' This equality does not follow from non-regularity together with almost 1-1; it only follows that the displayed set has positive measure. The argument still works with 'positive measure' instead of 'measure 1' (using continuity from below to find ε>0 with ν_eq(A_ε)>0), so the proof is repairable, but as written this is a false step.","section":"Proposition 4.11"}],"minor_comments":[{"comment":"In the definition of ε-stable in the mean, the expression 'sup_{n∈N} {1/N ∑_{i=1}^N ...}' mixes n and N; it should read 'sup_{N∈N} {1/N ∑_{i=1}^N ...}'.","section":"Definition 4.2"},{"comment":"The step 'By uniform ergodicity' for the indicator 1_{G_ε} is correct but deserves a justification, since 1_{G_ε} is not continuous. One can approximate 1_{G_ε} from below by continuous functions and use uniform convergence of their ergodic averages; please add this argument.","section":"Proposition 4.9"},{"comment":"In the claim, 'x_n ∈ π^{-1}_eq(Bη(y_n))' appears to be a typo for 'x_n ∈ π^{-1}_eq(y_n)'.","section":"Theorem 3.6 proof"},{"comment":"The companion assertion that a minimal almost diam-mean equicontinuous system is always diam-mean equicontinuous is also stated without proof; it should be justified or referenced.","section":"Section 4.1 (after Definition 4.6)"},{"comment":"The assertion that the multiple ergodic limit is positive by Furstenberg's theorem should be stated more precisely, with a reference to the specific result (e.g., positive multiple recurrence for distal systems).","section":"Section 6 (Theorem 6.1)"}],"recommendation":"major_revision","confidential_remarks":"This is a solid paper that is likely to be accepted after revisions. The two major issues are local: a missing proof for a transitive characterization used in Section 5, and a repairable false measure assertion in Proposition 4.11. The concern about the uniform ergodicity step in Proposition 4.9 is not valid, as that step can be justified by a standard approximation argument. The paper fits the scope of the journal and makes significant contributions to the hierarchy of strictly ergodic discrete-spectrum systems."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper does two genuinely new things. It introduces frequent stability and characterizes when the maximal equicontinuous factor map of a minimal mean equicontinuous system is almost automorphic (Theorem 3.6). And it fills the regularity rung by showing a minimal system is diam-mean equicontinuous iff its m.e.f. is regular (Theorem 4.12). The two equivalences are new and the overall hierarchy picture is clean. The positive-entropy transitive example and the partial answer to Furstenberg's question are nice bonuses.\n\nThe proofs are mostly coherent and use standard tools. That said, there are real soft spots. The most load-bearing is in Proposition 4.9: after constructing the open set Gε with ν_eq(Gε)>1-ε, the proof says 'by uniform ergodicity' the long blocks of the orbit of any y visit Gε with frequency ≥1-ε. Uniform ergodicity applies to continuous functions, or to Riemann-integrable functions whose discontinuity set has zero measure. The indicator of Gε is discontinuous on ∂Gε, and the proof never checks that ∂Gε is Haar-null. This matters because the bound on visits is what forces all long iterates of a small ball to have small diameter. The gap is likely repairable—choose the finite subcover radii so the sphere boundaries are null, or use a smooth approximation—but as written the direction (3)⇒(2) in Theorem 4.12 is not fully proved.\n\nSecond, the statement right after Definition 4.6 that a transitive system is almost Banach diam-mean equicontinuous iff it has one such point is asserted without proof. It is used in the proof of Theorem 5.6 to get residual equicontinuity points from a single one. That is not obvious for non-minimal transitive systems and needs a justification; otherwise the positive-entropy example may only exhibit one point.\n\nThird, Proposition 4.11 contains an overstrong measure statement: after assuming π_eq is not regular, it says the set with positive-diameter fibers has measure 1. That need not follow; the complement (singleton fibers) could have positive measure. The proof only needs positive measure, so this is a minor fix.\n\nFourth, the displayed word in Example 5.5 does not appear consistent with the recursion An+1 = An 0^{kn} Bn An. Probably a typo, but it makes the construction hard to follow.\n\nAll of this is fixable. The central claims are plausible and the new definitions are doing real work. This deserves a serious referee. I would recommend conditional acceptance after the gaps are closed. It is a good reading-group paper: the hierarchy is central and the errors are instructive. I'd likely cite it once the details are patched.\n\nRegards,","headline":"Fills two missing rungs in the mean-equicontinuity hierarchy, but the regularity theorem has a repairable proof gap.","tokens_in":16830,"tokens_out":3596,"would_cite":true,"duration_ms":33561,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["54H20"],"pacs":[],"model":"deepseek-v4-flash","headline":"For minimal systems, diam-mean equicontinuity, Banach diam-mean equicontinuity, and regularity of the maximal equicontinuous factor are equivalent, completing the discrete-spectrum hierarchy.","keywords":["mean equicontinuity","diam-mean equicontinuity","maximal equicontinuous factor","regularity","almost automorphic","discrete spectrum","strictly ergodic","topological entropy"],"falsifier":"Compute, for a minimal system whose maximal equicontinuous factor is known to be non-regular, the upper Banach density of the set of times at which the diameter of a small ball exceeds a fixed $\\varepsilon$. If this density can be made arbitrarily small by shrinking the ball, the claimed equivalence between regularity and Banach diam-mean equicontinuity fails; the theorem predicts a positive lower bound.","tokens_in":15824,"feed_emoji":"🎯","tokens_out":8860,"duration_ms":77447,"temperature":0.7,"pith_summary":"This paper characterizes two rungs in the hierarchy of strictly ergodic topological dynamical systems with discrete spectrum. It proves that a minimal system is diam-mean equicontinuous exactly when its maximal equicontinuous factor map is regular, meaning the points with a single preimage carry full Haar measure; the same theorem shows diam-mean and Banach diam-mean equicontinuity coincide in the minimal case. It also characterizes almost automorphic mean equicontinuous systems through a new local property, frequent stability, and completes the ladder from equicontinuity down to mere discrete spectrum. A constructed example shows that outside minimality, almost diam-mean equicontinuity allows positive topological entropy, and a partial answer to Furstenberg's multiple recurrence question is given for mean equicontinuous systems.","feed_headline":"Diam-mean equicontinuity equals a regular maximal factor","feed_subtitle":"For minimal systems, diam-mean, Banach diam-mean, and regular maximal factor are the same property.","key_machinery":"The central object is the maximal equicontinuous factor map $\\pi_{\\mathrm{eq}}\\colon X\\to X_{\\mathrm{eq}}$, the universal equicontinuous quotient of the system; its regularity, meaning singleton fibers on a full Haar-measure set, is the property being characterized. The main mechanism is the equivalence between local 'diam-mean' smallness of ball diameters and the measurable triviality of the $\\pi_{\\mathrm{eq}}$ fibers, proved by passing through the Besicovitch pseudometric $\\rho_b(x,y)=\\limsup_{n\\to\\infty}\\frac{1}{n}\\sum_{i=1}^n d(T^ix,T^iy)$, which vanishes exactly on pairs with the same maximal-factor image in minimal mean equicontinuous systems. A second mechanism is frequent stability: a point is frequently stable if small balls around it have diameter larger than $\\varepsilon$ only on a set of times of density less than $1$, and this property bridges mean equicontinuity with almost automorphy.","core_discovery":"The central discovery is Theorem 4.12: for a minimal topological dynamical system $(X,T)$, the following are equivalent: (1) $(X,T)$ is diam-mean equicontinuous; (2) $(X,T)$ is Banach diam-mean equicontinuous; (3) the maximal equicontinuous factor map $\\pi_{\\mathrm{eq}}\\colon X\\to X_{\\mathrm{eq}}$ is regular, i.e. the set $\\{y\\in X_{\\mathrm{eq}} : |\\pi_{\\mathrm{eq}}^{-1}(y)|=1\\}$ has full Haar measure $\\nu_{\\mathrm{eq}}$. Combined with known results, this yields the hierarchy: equicontinuity, nullness, tameness, diam-mean equicontinuity (regular maximal factor), mean equicontinuity plus frequent stability (almost one-to-one isomorphic maximal factor), mean equicontinuity (isomorphic maximal factor), and $\\mu$-mean equicontinuity (discrete spectrum). The paper also proves that for minimal mean equicontinuous systems the maximal factor map is almost one-to-one if and only if the system is frequently stable, and it constructs a transitive almost diam-mean equicontinuous system with positive topological entropy, separating local from global and Banach from non-Banach versions of these properties.","pith_inferences":["If the unproved transitive equivalence fails (almost diam-mean equicontinuity iff one diam-mean equicontinuity point), the positive-entropy example would only certify a single equicontinuity point; a direct check of whether the residual-set assertion in Theorem 5.6 uses that equivalence would settle the matter.","The regularity characterization suggests a plausible route for non-minimal systems: the equivalence in Theorem 4.12 may hold for transitive systems when the maximal factor is uniquely ergodic, analogous to the known mean equicontinuity case.","The hierarchy places diam-mean equicontinuity strictly between tameness and mean equicontinuity; this predicts that any minimal system with a regular maximal factor but not tame would separate tameness from diam-mean equicontinuity, should such a system exist.","The Furstenberg-type result for mean equicontinuous systems may extend to all systems whose maximal equicontinuous factor is regular and whose unique invariant measure makes the system measurably distal, since the argument uses only the measurable isomorphism and pointwise multiple ergodic averages."],"forward_implications":["Every rung of the hierarchy for strictly ergodic discrete-spectrum systems is now characterized: equicontinuity, nullness, tameness, diam-mean equicontinuity, mean equicontinuity with frequent stability, mean equicontinuity, and $\\mu$-mean equicontinuity.","Every minimal tame system is Banach diam-mean equicontinuous, since its maximal equicontinuous factor is regular.","In minimal systems, diam-mean equicontinuity and Banach diam-mean equicontinuity coincide; the paper's example shows they differ locally for transitive systems.","There exists a transitive almost diam-mean equicontinuous system with positive topological entropy, showing this class is wider than the zero-entropy classes.","For a minimal mean equicontinuous system, almost every point with respect to the unique invariant measure is a minimal point for the product system $T\\times T^2\\times\\cdots\\times T^d$ for every $d$, giving a partial answer to Furstenberg's question."],"supporting_citations":[{"why":"Supplies the result that a minimal system is mean equicontinuous if and only if the maximal equicontinuous factor map is isomorphic, forming the foundation for the hierarchy and for Theorem 3.6.","marker":"[28]"},{"why":"Provides the converse direction of the isomorphic-factor characterization and the almost-one-to-one dichotomy lemma used in Theorem 3.6, and constructs an example where the maximal factor is not almost one-to-one.","marker":"[6]"},{"why":"Introduced diam-mean equicontinuity and showed minimal null systems are diam-mean equicontinuous; the dichotomy between diam-mean equicontinuity and diam-mean sensitivity is taken from there.","marker":"[14]"},{"why":"Establishes that the maximal equicontinuous factor of every minimal tame system is regular, yielding Corollary 4.10.","marker":"[11]"},{"why":"Proves mean equicontinuity equals Banach mean equicontinuity for non-minimal systems, used in the discussion after Theorem 4.12 and in Corollary 5.8.","marker":"[29]"},{"why":"Provides the pointwise convergence of multiple ergodic averages that powers the Furstenberg-type result in Theorem 6.1.","marker":"[23]"},{"why":"Furstenberg's question on minimal points for $T\\times T^2\\times\\cdots\\times T^d$, which Theorem 6.1 answers for mean equicontinuous systems.","marker":"[10]"}],"fun_headline_variants":["For minimal systems, diam-mean = Banach diam-mean = regular factor","Regular maximal factor iff diam-mean equicontinuity for minimal","Regular factor characterizes diam-mean equicontinuity in minimal systems","Diam-mean, Banach diam-mean, regular factor: all same for minimal","Diam-mean equals regular factor, closing the hierarchy"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument relies on the unproved claim that for transitive (not necessarily minimal) systems, having a single diam-mean equicontinuity point already forces the set of such points to be residual; if that claim fails, the positive-entropy example only demonstrates a single point, not the advertised dense set.","fun_headline_variants_meta":{"raw":{"variants":["For minimal systems, diam-mean = Banach diam-mean = regular factor","Regular maximal factor iff diam-mean equicontinuity for minimal","Regular factor characterizes diam-mean equicontinuity in minimal systems","Diam-mean, Banach diam-mean, regular factor: all same for minimal","Diam-mean equals regular factor, closing the hierarchy"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00135,"raw_usage":{"total_tokens":5510,"prompt_tokens":1003,"completion_tokens":4507,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":619,"completion_tokens_details":{"reasoning_tokens":4408}},"tokens_in":619,"tokens_out":4507,"duration_ms":30968,"temperature":1.0,"reasoning_tokens":4408,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:22:08.199983+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for a minimal system whose maximal equicontinuous factor is known to be non-regular, the upper Banach density of the set of times at which the diameter of a small ball exceeds a fixed $\\varepsilon$. If this density can be made arbitrarily small by shrinking the ball, the claimed equivalence between regularity and Banach diam-mean equicontinuity fails; the theorem predicts a positive lower bound.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the result that a minimal system is mean equicontinuous if and only if the maximal equicontinuous factor map is isomorphic, forming the foundation for the hierarchy and for Theorem 3.6."},{"cited_title":"Downarowicz and E","cited_arxiv_id":null,"evidence_quote":"Provides the converse direction of the isomorphic-factor characterization and the almost-one-to-one dichotomy lemma used in Theorem 3.6, and constructs an example where the maximal factor is not almost one-to-one."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduced diam-mean equicontinuity and showed minimal null systems are diam-mean equicontinuous; the dichotomy between diam-mean equicontinuity and diam-mean sensitivity is taken from there."},{"cited_title":"Qiu and J","cited_arxiv_id":null,"evidence_quote":"Proves mean equicontinuity equals Banach mean equicontinuity for non-minimal systems, used in the discussion after Theorem 4.12 and in Corollary 5.8."},{"cited_title":"Pointwise convergence of multiple ergodic averages and strictly ergodic models","cited_arxiv_id":"1406.5930","evidence_quote":"Provides the pointwise convergence of multiple ergodic averages that powers the Furstenberg-type result in Theorem 6.1."},{"cited_title":"Furstenberg, Poincare recurrence and number theory, Bull","cited_arxiv_id":null,"evidence_quote":"Furstenberg's question on minimal points for $T\\times T^2\\times\\cdots\\times T^d$, which Theorem 6.1 answers for mean equicontinuous systems."}],"review_version":1}