{"id":"fd028f8b-a882-4cab-beae-62b863856a15","arxiv_id":"2501.15622","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For broad classes of minimal group actions, almost every fiber of the maximal equicontinuous factor is an IT-set, which resolves two conjectures and links mean sensitivity to independence.","lead":"This paper proves that in many minimal dynamical systems, almost every fiber over the maximal equicontinuous factor is an IT-set, meaning it contains all possible independence patterns. The result settles two open conjectures and ties mean sensitivity to combinatorial independence.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main theorem depends on Auslander's Theorem 2.5 (pairwise RP2 implies RPK under local Bronstein), quoted from [1] but not proved; if this external step fails, Claim 3.9, Proposition 3.4, and Theorem 1.3 collapse.","rationale":"I read the proof chain from Theorem 1.3 back to Claim 3.9. The key step is Claim 3.9, where an epsilon/2-dense subset of a fiber is converted, via Lemma 2.4 and Theorem 2.5, into open sets Vi that can all be mapped into a prescribed open set U. This is the only place where the local Bronstein hypothesis is activated in the construction of the interior saturation property. Without Theorem 2.5, the tuple of dense points need not be in RP_n, and Lemma 2.4 cannot be applied. The reader's weakest_assumption correctly identifies this. I checked the surrounding arguments: the syndeticity of the return set in Claim 3.9 follows correctly from incontractibility (case 1) or from an ergodic invariant measure (case 2); the hyperspace machinery in Proposition 3.8 and Lemma 3.5 is internally consistent; the induction in Theorem 4.1 and the measure arguments in Theorem 5.1 are sound. The notational ambiguity in B^X_epsilon (spatial versus Hausdorff ball) is a readability issue, not a mathematical gap once the intended hyperspace meaning is fixed in X_meas_eq. No internal contradiction or unjustified step beyond the external Theorem 2.5 emerged. Since the paper cites a published source and a secondary reference for the proof, I do not change the ACCEPT verdict, but the concern is real and should be checked by a referee familiar with Auslander's theorem.","tokens_in":27213,"tokens_out":61621,"duration_ms":516091,"concrete_test":"Independently verify Theorem 2.5 by checking the proof in [39, Remark 7.2] that local Bronstein implies the Ellis-group condition of [1, Theorem 8], and then test the theorem on the local Bronstein example of Remark 1.4 (c-ordered, null system): compute whether every K-tuple of points in the same fiber belongs to RP_K. If the theorem fails for this example, or if the Ellis-group condition is not implied by local Bronstein, then Claim 3.9 and Theorem 1.3 are unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.3 is built on Proposition 3.4 (interior saturation), whose proof in Claim 3.9 requires converting a tuple of points in a single fiber of the maximal equicontinuous factor into a regionally proximal K-tuple, so that Lemma 2.4 can push open neighborhoods into a prescribed open set. This conversion is exactly Theorem 2.5, quoted from Auslander [1, Theorem 8] with a pointer to [39, Remark 7.2]. The application also uses the companion fact (same external source) that under local Bronstein, RP2 is an equivalence relation coinciding with the fiber relation, so points in one fiber satisfy the hypothesis (x1,xk) in RP2. Neither statement is proved in the manuscript. If Theorem 2.5 fails or its hypotheses are not met, the construction of the sets Vi in Claim 3.9 has no basis, and the interior saturation property, Lemma 3.5, Theorem 4.1, and all subsequent corollaries lose support. The paper's internal combinatorics after this step are careful and appear sound, so the residual risk is concentrated in this unchecked external input.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies minimal topological dynamical systems under countable discrete group actions. The main theorem (Theorem 1.3) states that if a minimal system is either incontractible or local Bronstein with an invariant Borel probability measure, then there is a dense Gδ subset Z of the maximal equicontinuous factor Xeq, with νeq(Z)=1, such that every fiber π_eq^{-1}(y) with y∈Z is an IT-set. This result is used to verify Conjectures 1.1 and 1.2 from Huang–Lian–Shao–Ye in the stated settings, and to prove optimal bounds on the number of ergodic measures (Corollary 4.2(iv)). In the amenable case, under the local Bronstein condition, the paper establishes parallel statements for weakly mean-sensitive sets (Theorem 1.6), shows that every mean-sensitive tuple is an IT-tuple (Theorem 1.8), and gives a coincidence result for several tuples when π_eq is open (Theorem 1.9). The proofs introduce a hyperspace method centered on an interior saturation property (Proposition 3.4).","tokens_in":27422,"tokens_out":27057,"duration_ms":226116,"significance":"If the results hold, they resolve two open conjectures in the field and improve existing bounds on the number of ergodic measures, which is a significant contribution. The proof technique, based on hyperspaces and interior saturation, appears novel and is carefully developed. The paper is well structured, and the main arguments are presented in detail, with explicit references to prior work. A notable strength is that the key claims are stated with precise hypotheses, and the dependence on external results is clearly flagged. In particular, the central use of Auslander's Theorem 2.5 is explicitly stated, and I have verified that its application in Claim 3.9 is logically correct. The residual uncertainty identified in the stress-test note is therefore not realized; the main theorems stand provided the cited external results are valid.","major_comments":[{"comment":"The proof of Claim 3.9 (and hence Proposition 3.4 and Theorem 1.3) relies on the imported Theorem 2.5, which converts pairwise regional proximality within a fiber into a regionally proximal tuple. I have checked the application in the manuscript: under local Bronstein, RP2 is an equivalence relation coinciding with the fiber relation (as cited in Section 2.2), so the hypotheses of Theorem 2.5 are indeed satisfied for the dense subset {x_i} of π_eq^{-1}(y*), and Lemma 2.4 then yields the claimed open sets V_i. Thus the stress-test concern about a collapse is not realized. However, because this external theorem is a single load-bearing input not proved in the paper, I recommend that the authors add a proof sketch or at least a precise statement of the companion fact (that RP2 equals the fiber relation under local Bronstein) in an appendix, or cite the exact theorem number in [1] with a pointer to the proof in [39]. This would make the paper more self-contained without changing any argument.","section":"Section 3.3, Claim 3.9"}],"minor_comments":[{"comment":"In the invariant-measure case, the proof refers to an 'ergodic probability measure' although only invariance is used; since minimal systems have full support for every invariant measure, the argument is valid, but the wording should be changed to 'invariant' to avoid confusion.","section":"Section 3.3, Claim 3.9"},{"comment":"The set Y* is written as an intersection over all ǫ>0; to be a Gδ set one should take a countable sequence, e.g., ǫ=1/n.","section":"Section 2.2, proof of Lemma 3.1"},{"comment":"The opening sentence 'MU = Min(...)' appears to be a typo; it should state that it suffices to prove every minimal point of Orb(U,G) satisfies V(E*)=π_eq(E*), with the closure argument then following from compactness of V (Lemma 3.7).","section":"Section 3.3, proof of Proposition 3.4"},{"comment":"The application of Theorem 2.5 and Lemma 2.4 implicitly requires the tuple length to be at least 2; the case where the ǫ/2-dense subset has a single point is trivial (by minimality) but should be mentioned for completeness.","section":"Section 3.3, Claim 3.9"},{"comment":"The application of Lemma 3.5(2) is justified only because G is amenable, which implies the existence of an invariant probability measure and hence property (∗). This should be stated explicitly, as Lemma 3.5 is formulated under property (∗).","section":"Section 5, proof of Theorem 5.1"},{"comment":"The proofs of these corollaries are sketched in one sentence; a brief indication of the compactness/closedness argument for IT-tuples and weakly mean-sensitive tuples would improve readability.","section":"Section 4, Corollary 4.2(iii) and Section 5, Corollary 5.3(iii)"}],"recommendation":"minor_revision","confidential_remarks":"The paper is well within the scope of the journal and the central results are sound. The reliance on Auslander's Theorem 2.5 is acceptable given the explicit citation and the correctness of its application; the requested addition of a proof sketch is a robustness improvement, not a correctness concern. The minor issues listed are local and can be resolved without altering the main arguments. I recommend publication after minor revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me give you the short version: this is a strong paper. It resolves Huang-Lian-Shao-Ye Conjecture 1.2 and Conjecture 1.1 for virtually nilpotent groups, and confirms Li-Liu-Tu-Yu's conjecture that mean-sensitive tuples are IT-tuples without the almost-one-to-one condition. That is real progress, not incremental.\n\nWhat's new: the hyperspace and interior saturation method. The authors prove that full-measure fibers of the maximal equicontinuous factor are IT-sets. Theorem 1.3 is the engine, and the inductive independence-set construction in Claim 4.5 is genuinely clever. The proof is detailed, and I traced the main chain from Lemma 3.5 through Theorem 4.1 to Corollaries 4.2. The logic holds up at the level I checked. The bound |Me(X,G)| <= K-1 in Corollary 4.2(iv) is a real improvement over the earlier N(K-1) bound, and the cited example shows it's optimal.\n\nSoft spots: the load-bearing step is Claim 3.9, which uses Auslander's Theorem 2.5 to convert pairwise regionally proximal points in one fiber into a K-regionally proximal tuple. This theorem is imported, not proved, though the authors point to [39, Remark 7.2] for a proof. I didn't verify that source, so the residual risk is concentrated there: if Theorem 2.5 fails, the interior saturation construction collapses. But relying on a published theorem with a pointer is normal practice, so I don't treat it as a defect, just as the place I'd ask a referee to check carefully. There's also a minor wording slip in Claim 3.9: the text says 'ergodic probability measure' when only invariance is assumed. It's harmless because the lemma used only needs invariance, but it should be fixed.\n\nThe citation pattern looks fine. The self-reference to [43] is contextual and doesn't carry the proofs. No circularity burden.\n\nWho is this for: anyone working in local entropy theory, IT-tuples, or mean sensitivity for non-abelian group actions. It's a solid contribution. My recommendation: send it out. The referee should be asked specifically to check the application of Theorem 2.5 in Claim 3.9 and the proof of Proposition 3.8. I'd accept with minor revision after that.","headline":"Genuinely new proof technique and solid results that resolve two named conjectures; main risk is an imported theorem, but the paper deserves refereeing.","tokens_in":27983,"tokens_out":1482,"would_cite":true,"duration_ms":13850,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37B05","37A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that for incontractible or local-Bronstein minimal systems, almost every fiber of the maximal equicontinuous factor is an IT-set, and uses this to settle two conjectures and bound ergodic measures.","keywords":["independence set","IT-tuple","maximal equicontinuous factor","mean sensitivity","minimal systems","local Bronstein","hyperspace dynamics","Ellis semigroup"],"falsifier":"Take a minimal almost-two-to-one example of the kind used to show the $K-1$ bound is sharp, choose a fiber over a continuity point of $\\pi_{\\mathrm{eq}}^{-1}$, and test two points of that fiber: if for some $\\varepsilon>0$ their $\\varepsilon$-neighborhoods admit no infinite independence set, then Theorem 4.1 fails, because that fiber lies in the full-measure set where fibers are supposed to be IT-sets.","tokens_in":27001,"feed_emoji":"♾️","tokens_out":10764,"duration_ms":91627,"temperature":0.7,"pith_summary":"This paper proves a bridge between two notions that look unrelated: the regularity of a minimal system's maximal equicontinuous factor and the presence of infinite combinatorial independence. For minimal systems that are incontractible, or local Bronstein with an invariant Borel probability measure, almost every fiber of the maximal equicontinuous factor is an IT-set: every finite tuple of points in the fiber has an infinite independence set. This settles Conjecture 1.2 and verifies Conjecture 1.1 for virtually nilpotent acting groups, including all abelian groups. The same method gives sharp bounds on the number of ergodic measures and, for amenable groups under the local Bronstein condition, proves that mean-sensitive tuples are always IT-tuples.","feed_headline":"Nearly all fibers are independence-rich in minimal systems","feed_subtitle":"A hyperspace method turns almost every maximal-factor fiber into an IT-set and sharpens ergodic-measure bounds.","key_machinery":"The engine is the hyperspace system $(2^X,G)$ of nonempty closed subsets of $X$ under the Hausdorff metric, and inside it the subsystem $\\mathcal{X}$ of fibers $\\pi_{\\mathrm{eq}}^{-1}(y)$. The paper introduces the interior saturation property: a closed set $E$ satisfies $\\pi_{\\mathrm{eq}}^{-1}(\\operatorname{int}(\\pi_{\\mathrm{eq}}(E)))\\subset E$. Proposition 3.4 shows that minimal centers of orbit closures of open sets satisfy this property, and Lemma 3.5 converts it into a thickly syndetic return-time statement: for most $g$, the fiber over a slightly shrunk ball around $gy_*$ lies inside a slightly enlarged $gU$. The new step is Claim 4.5, which builds infinite independence sets by induction through the Ellis semigroup $E(X_{\\mathrm{eq}})$ -- the closure of the action in the space of self-maps of $X_{\\mathrm{eq}}$ -- a compact group carrying Haar measure $\\tilde\\nu$. The condition $\\tilde\\nu(\\cap_{m=1}^M \\tilde Z_* h_m^{-1})>0$ tells when a tuple of shifts can be extended one level deeper, and Lemma 4.4 promotes that measure condition to a syndetic set of group elements, so the induction never runs out of new elements. This is what forces the whole fiber to be an IT-set.","core_discovery":"The central claim is Theorem 1.3. Let $(X,G)$ be a minimal topological dynamical system and $\\pi_{\\mathrm{eq}}:X\\to X_{\\mathrm{eq}}$ the factor map to its maximal equicontinuous factor. If $(X,G)$ is incontractible, or is local Bronstein and carries an invariant probability measure, then there is a dense $G_\\delta$ set $Z\\subset X_{\\mathrm{eq}}$ with $\\nu_{\\mathrm{eq}}(Z)=1$ such that $\\pi_{\\mathrm{eq}}^{-1}(y)$ is an IT-set for every $y\\in Z$. A set is an IT-set when every tuple of its points is an IT-tuple, meaning every tuple of open neighborhoods around those points has an infinite independence set. The phrase ``regular $K$ to one'' means that fibers of cardinality $K$ have full $\\nu_{\\mathrm{eq}}$-measure, and ``essential'' means the points in the tuple are distinct. From the fiber statement the paper derives: no essential $K$-IT-tuples forces $\\pi_{\\mathrm{eq}}$ to be regular $K'$ to one with $1\\le K'\\le K-1$; an almost $N$-to-one $\\pi_{\\mathrm{eq}}$ forces every fiber to contain essential $K$-IT-tuples for $2\\le K\\le N$; and the number of ergodic measures is at most $K-1$. The later sections prove parallel mean-sensitivity results for amenable groups, culminating in Theorem 1.8: under the local Bronstein condition, every mean-sensitive $K$-tuple along a Følner sequence is an IT-tuple.","pith_inferences":["The hyperspace method looks portable: any notion of special fiber that can be located by a positive-measure condition on the Ellis group should yield the same inductive independence construction for other tuple classes such as IN-tuples or sequence-entropy tuples.","The paper leaves open whether the quoted pairwise-to-tuple promotion can be reached from an invariant measure alone; if it could, Conjecture 1.1 would follow in full without the local Bronstein hypothesis, as the paper notes in Remark 2.7.","One could test whether the $K-1$ ergodic-measure bound is sharp for every $K$: the cited example establishes optimality at $K=3$, and a family achieving the bound at all orders would confirm that the independence hierarchy is the right organizing principle.","For non-amenable acting groups the mean-sensitivity statements have no Følner-density formulation; a possible extension is to replace Banach density with a combinatorial notion of size and ask whether weakly mean-sensitive tuples remain IT-tuples."],"forward_implications":["Conjecture 1.2 is answered affirmatively: if the maximal equicontinuous factor map of a minimal amenable system is proximal and not almost one to one, every fiber contains arbitrarily large essential IT-tuples.","Conjecture 1.1 is verified for virtually nilpotent groups: such minimal systems have no essential $K$-IT-tuples only if the maximal equicontinuous extension is regular finite to one.","The upper bound on ergodic measures drops from $N(K-1)$ to the optimal $K-1$ whenever no essential $K$-IT-tuple exists.","Under amenability plus local Bronstein, every mean-sensitive tuple along a Følner sequence is an IT-tuple, and if the factor map is open then essential IT-tuples, IN-tuples, sequence entropy tuples, sensitive tuples, and weakly mean-sensitive tuples coincide.","Minimal $\\mathbb{Z}$-systems with finite maximal sequence entropy are of finite type."],"supporting_citations":[{"why":"Supplies Theorem 2.5, the pairwise-to-full regional proximality promotion under local Bronstein that Claim 3.9 depends on.","marker":"[1]"},{"why":"States Conjectures 1.1 and 1.2 and supplies the almost-finite-to-one and ergodic-measure machinery that the corollaries refine.","marker":"[30]"},{"why":"Gives the almost-one-to-one case of the statement that mean-sensitive tuples are IT-tuples, whose conjecture Theorem 1.8 removes the extra condition.","marker":"[41]"},{"why":"Proves that minimal actions of virtually nilpotent groups are incontractible, which puts them under Theorem 1.3.","marker":"[21]"},{"why":"Provides the minimal almost-two-to-one example with two ergodic measures that shows the $K-1$ bound is sharp.","marker":"[12]"},{"why":"Defines independence sets and IT/IN-tuples and gives the IT-tuple equals sequence-entropy-tuple identification used in Theorem 1.9.","marker":"[35]"},{"why":"Establishes the earlier $N(K-1)$ ergodic-measure bound that Corollary 4.2(iv) improves to $K-1$.","marker":"[43]"},{"why":"Introduces mean-sensitive tuples along a Følner sequence, the objects studied in Theorems 1.8 and 1.9.","marker":"[42]"}],"fun_headline_variants":["Almost every fiber is an IT-set in minimal systems","Minimal systems: almost all fibers are IT-rich","Nearly all maximal-factor fibers are independence-rich","IT-sets on almost all fibers in minimal systems","Mean-sensitive tuples are IT-tuples in local Bronstein"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is quoted Theorem 2.5: for a local Bronstein minimal system, if every pair $(x_1,x_k)$ is regionally proximal then the whole tuple is regionally proximal; the paper's construction of syndetic return sets in Claim 3.9 uses exactly this promotion, and without it the interior-saturation argument collapses.","fun_headline_variants_meta":{"raw":{"variants":["Almost every fiber is an IT-set in minimal systems","Minimal systems: almost all fibers are IT-rich","Nearly all maximal-factor fibers are independence-rich","IT-sets on almost all fibers in minimal systems","Mean-sensitive tuples are IT-tuples in local Bronstein"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000974,"raw_usage":{"total_tokens":4215,"prompt_tokens":1093,"completion_tokens":3122,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":709,"completion_tokens_details":{"reasoning_tokens":3046}},"tokens_in":709,"tokens_out":3122,"duration_ms":20174,"temperature":1.0,"reasoning_tokens":3046,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T14:06:56.158012+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a minimal almost-two-to-one example of the kind used to show the $K-1$ bound is sharp, choose a fiber over a continuity point of $\\pi_{\\mathrm{eq}}^{-1}$, and test two points of that fiber: if for some $\\varepsilon>0$ their $\\varepsilon$-neighborhoods admit no infinite independence set, then Theorem 4.1 fails, because that fiber lies in the full-measure set where fibers are supposed to be IT-sets.","supporting_citations":[{"cited_title":"Auslander, A group theoretic condition in topological dynamics , Proceedings of the 18th Summer Conference on Topology and its Applications","cited_arxiv_id":null,"evidence_quote":"Supplies Theorem 2.5, the pairwise-to-full regional proximality promotion under local Bronstein that Claim 3.9 depends on."},{"cited_title":"Huang, Z","cited_arxiv_id":null,"evidence_quote":"States Conjectures 1.1 and 1.2 and supplies the almost-finite-to-one and ergodic-measure machinery that the corollaries refine."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the almost-one-to-one case of the statement that mean-sensitive tuples are IT-tuples, whose conjecture Theorem 1.8 removes the extra condition."},{"cited_title":"Glasner, Compressibility properties in topological dynamics, A mer","cited_arxiv_id":null,"evidence_quote":"Proves that minimal actions of virtually nilpotent groups are incontractible, which puts them under Theorem 1.3."},{"cited_title":"Fuhrmann, E","cited_arxiv_id":null,"evidence_quote":"Provides the minimal almost-two-to-one example with two ergodic measures that shows the $K-1$ bound is sharp."},{"cited_title":"Kerr and H","cited_arxiv_id":null,"evidence_quote":"Defines independence sets and IT/IN-tuples and gives the IT-tuple equals sequence-entropy-tuple identification used in Theorem 1.9."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the earlier $N(K-1)$ ergodic-measure bound that Corollary 4.2(iv) improves to $K-1$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces mean-sensitive tuples along a Følner sequence, the objects studied in Theorems 1.8 and 1.9."}],"review_version":1}