{"id":"28d3ecf7-bf58-45fb-8edc-ba7a5a04aca2","arxiv_id":"2501.08069","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For invariant measures on countable group actions, every sequence entropy tuple is an IT tuple, and for amenable groups these coincide with mean sensitive tuples.","lead":"The paper proves that every measure-theoretic sequence entropy tuple is an independence (IT) tuple for any countable infinite discrete group action, and that for amenable group actions these tuples coincide with mean sensitivity tuples. It generalizes results previously known only for integer or abelian group actions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.6 proof omits the positive-measure argument that an ergodic z satisfying the shifted intersection and positivity conditions exists; the step is fixable but is the true load-bearing point.","rationale":"The reader correctly located the fragile step in Section 5.2, but the precise difficulty is not that Lemma 5.6 is inapplicable: with z=(e,…,e), Lemma 5.6 does apply because λ^X_K=λ^X_{(e,…,e)}. The load-bearing gap is the existence of one z satisfying all three conditions simultaneously. The diagonal choice gives (2) and (3) but not guaranteed ergodicity, while the full-measure ergodic set from Lemma 5.4 need not meet that diagonal. The required positive-measure intersection argument is missing. I verified that the argument can be supplied, so the concern is fixable rather than fatal. The other results, especially Theorems 1.1 and 1.3, appear sound, and Theorem 1.6's other inclusion SM⊂SE is routine. Thus the manuscript should not be rejected, but the authors should be required to fill the omitted argument before the headline equality is accepted. This is consistent with the reader's CONDITIONAL verdict; no verdict change is needed.","tokens_in":18283,"tokens_out":26208,"duration_ms":267958,"concrete_test":"Supply the missing paragraph in Section 5.2 after Lemma 2.5: define S={z∈W^K:ν_Z(∩Z_V z_k^{-1})>0} and prove ν_Z^K(S)>0 via E[ν_Z(∩Z_V z_k^{-1})]=∫_Z (ν_Z(w^{-1}Z_V∩W)/ν_Z(W))^K dν_Z(w)>0, then intersect S with the full-measure ergodic set from Lemma 5.4. If this computation cannot be completed as written, Theorem 1.6's proof has a genuine gap in the SE-to-MS inclusion.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 5.2, after obtaining λ^X_K(U_1×⋯×U_K)=2b>0, the proof states 'By Lemma 5.6, ν_Z(∩Z^b_{U_k})>b'. This is valid only with the unstated choice z=(e,…,e), since λ^X_K=λ^X_{(e,…,e)}; the text does not spell out this identification. The more serious gap is the next sentence: 'together with Lemma 2.5 and Lemma 5.4, implies that there exists z∈Z^K' satisfying (1) λ^X_z ergodic, (2) ν_Z(∩Z^b_{U_k}z_k^{-1})>b, and (3) ν_Z(∩Z_V z_k^{-1})>0. Lemma 2.5 only gives an open neighborhood W of e such that (2) holds for every z_k∈W. Condition (3) holds for z_k=e, but z=(e,…,e) need not be in the full-measure ergodic set from Lemma 5.4; the diagonal has measure zero in general. The existence of an ergodic z in W^K with (3) requires a separate argument. One can supply it: the set S={z∈W^K:ν_Z(∩Z_V z_k^{-1})>0} has positive ν_Z^K-measure, because its expected intersection measure equals ∫_Z (ν_Z(w^{-1}Z_V∩W)/ν_Z(W))^K dν_Z(w)>0, using ν_Z(Z_V)>0. Intersecting S with the full-measure ergodic set from Lemma 5.4 yields the desired z. This argument is absent from the manuscript, so the SE^μ_K⊂MS^μ_K inclusion is not fully justified as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies local entropy theory for actions of countable infinite discrete groups. Theorem 1.1 shows that every measure-theoretic sequence entropy K-tuple of an invariant measure is an IT K-tuple, yielding with known results the chain \\cup_\\mu SE^\\mu_K \\subset IT_K \\subset SE_K. Theorem 1.3 gives an upper bound log(K−1) on maximal sequence entropy when no essential sequence entropy K-tuples exist, and Corollary 1.5 extends this to absence of essential IT tuples. Theorem 1.6 claims that for ergodic measures of amenable group actions, sequence entropy tuples, mean sensitive tuples, and sensitive-in-the-mean tuples coincide along any tempered Følner sequence. The proofs of Theorems 1.1 and 1.3 are essentially sound; the proof of Theorem 1.6 contains a gap in the SE⊂MS inclusion concerning the existence of an ergodic shift z satisfying both a positivity condition from Lemma 2.5 and a support condition from the given positive-measure set.","tokens_in":18666,"tokens_out":18139,"duration_ms":151202,"significance":"If Theorem 1.6 can be repaired, the paper settles an open question from the local entropy theory literature by giving a complete local characterization of sequence entropy via mean sensitivity for ergodic measures under amenable group actions, going beyond the abelian case treated in [6] and [25]. The novelty also includes the IT-tuple result (Theorem 1.1) and the entropy bound (Theorem 1.3), which are clean and appear correct. The proof technique, using the Kronecker factor and disintegration measures, is elegant and likely portable. The gap in Theorem 1.6 is localized and repairable, so the paper's contribution is substantial despite the current lack of full justification.","major_comments":[{"comment":"In the proof of SE^\\mu_K ⊂ MS^\\mu_K, the step 'By Lemma 5.6, ν_Z(∩ Z^b_{U_k}) > b, which together with Lemma 2.5 and Lemma 5.4, implies that there exists z=(z_1,…,z_K)∈Z^K such that …' is not justified as written. To apply Lemma 5.6 to λ^X_K one must first identify λ^X_K with λ^X_z for z=(e,…,e); this identification is true (via (2.5) and (5.5)) but is never stated. More importantly, the existence of an ergodic z satisfying both conditions (2) and (3) does not follow from Lemmas 2.5 and 5.4 alone: Lemma 2.5 gives an open neighborhood W of e such that (2) holds for every z∈W^K, and Lemma 5.4 gives a full-measure set E of z for which λ^X_z is ergodic, but the point z=(e,…,e) need not belong to E, and the diagonal has ν_Z^{(K)}-measure zero. A missing positive-measure argument is needed, for instance showing that S={z∈W^K: ν_Z(∩ Z_V z_k^{-1})>0} has positive ν_Z^{(K)}-measure by computing ∫_{W^K} ν_Z(∩ Z_V z_k^{-1}) dν_Z^{(K)}(z) = ∫_Z (ν_Z(z^{-1}Z_V∩W))^K dν_Z(z) > 0, then intersecting S with E to obtain the desired z. The manuscript omits this argument, so the inclusion SE^\\mu_K ⊂ MS^\\mu_K is not fully proved as written. The gap is fixable, but it is load-bearing for Theorem 1.6.","section":"Section 5.2"}],"minor_comments":[{"comment":"In the proof of Theorem 1.3, the sentence 'We now suppose that (X,G) has essential sequence entropy K-tuples for µ' should read 'does not have essential sequence entropy K-tuples' to match the theorem statement and the subsequent use of Lemma 4.1.","section":"Section 4"},{"comment":"In the proof of Proposition 2.7, the phrase 'an open neighborhood of (x_1,…,x_K)' refers to the set X\\B_k, which is not necessarily open; it should be X\\setminus \\overline{B_k} throughout that argument.","section":"Appendix A"},{"comment":"The final line of the proof of Theorem 1.6 writes SM^\\mu_K(X,G) ⊂ SE^\\mu_K(X,T); the last argument should be SE^\\mu_K(X,G).","section":"Section 5.2"},{"comment":"The statement 'the µ-sequence entropy K-tuple, the µ-mean sensitive K-tuple along F and the µ-sensitive in the mean K-tuple along F coincide' should use the plural 'tuples'.","section":"Theorem 1.6"},{"comment":"The application of (5.11) requires a measurable set A⊂L with µ(A)>0 and diam(A)<δ'; such a set exists because µ(L)>0 and µ is a finite Borel measure on a compact metric space, but this elementary fact is not stated.","section":"Section 5.2, SM⊂SE step"},{"comment":"There are minor typos: 'supremun' should be 'supremum' in the definition of h^*_µ, and 'The later' should be 'The latter' in the proof of Lemma 5.4.","section":"Section 2.6 and Lemma 5.4"}],"recommendation":"major_revision","confidential_remarks":"The paper is a good fit for the journal. The gap in Theorem 1.6 is genuinely fixable, and the rest of the manuscript is in good shape. I recommend asking the authors to add the missing positive-measure argument in Section 5.2 and to correct the small presentation issues listed above. The reliance on the authors' own previous work for Theorem 2.6 is appropriate because that result is independently available in the literature (Kerr–Li, Huang–Maass–Ye); no concerns about the citation pattern otherwise."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a serious paper with real new content. Theorems 1.1 and 1.3 look correct; the proofs are honest adaptations of Kerr–Li and Huang–Ye, with proper sourcing. Theorem 1.3 usefully extends the sequence-entropy bound to countable discrete groups and gives the Corollary 1.5 about IT tuples. The genuinely new piece is Theorem 1.6, the coincidence of sequence entropy tuples, mean sensitive tuples, and sensitive-in-the-mean tuples for ergodic measures of amenable group actions. That is the result people will care about.\n\nThe proof of Theorem 1.6 has a gap, and the reader's concern is accurate. After Lemma 5.6 gives ν_Z(∩ Z^b_{U_k}) > b, the text jumps to the existence of an ergodic z satisfying the shifted intersection conditions. Two things are missing. First, the identification λ^X_K = λ^X_{(e,...,e)} is true but never stated; it follows from the disintegration over the Kronecker factor. Second, and more importantly, Lemma 5.4 only guarantees ergodicity of λ^X_z for ν_Z^K-almost every z. The diagonal point (e,...,e) need not lie in that full-measure set, and the condition ν_Z(∩ Z_V z_k^{-1}) > 0 holds at the diagonal but not automatically nearby. The stress-test note is right that this can be fixed: integrate over w to show the set of z satisfying the intersection condition has positive ν_Z^K-measure, then intersect with the ergodic full-measure set. That argument is absent from the manuscript, so the inclusion SE^μ_K ⊂ MS^μ_K is not fully justified as written.\n\nEverything else in the section looks fine. The second inclusion SM ⊂ SE uses standard almost-periodic function arguments and seems sound. The appendix proof of Proposition 2.7 is routine and correct. Citation practice is reasonable; Theorem 2.6 is cited from prior work and independently established, so no circularity problem.\n\nThis paper deserves a serious referee, not a desk reject. I would send it out and ask the authors to supply the missing z-existence argument and to spell out the diagonal identification. The gap is likely fixable and the surrounding results are convincing. If the fix works, the paper should be publishable.","headline":"Solid extension of local entropy theory to amenable groups, with a real but repairable gap in the proof of the headline Theorem 1.6.","tokens_in":19180,"tokens_out":1531,"would_cite":true,"duration_ms":16983,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37A15","37A35","37B05"],"pacs":[],"model":"deepseek-v4-flash","headline":"For an ergodic measure on an amenable group action, a tuple is a sequence-entropy tuple exactly when it is mean-sensitive along a tempered Følner sequence.","keywords":["IT tuple","sequence entropy tuple","mean sensitive tuple","amenable group action","Kronecker factor","tempered Følner sequence","ergodic measure","independence set"],"falsifier":"Compute, for an ergodic rotation on a compact abelian group (say the circle) and two small arcs, the diagonal measure of the product of the arcs and the Haar measure of the set of rotations for which both fibres have mass greater than $b$; the proof predicts the latter exceeds $b$ whenever the former exceeds $2b$. A concrete example violating this inequality would show that the step from diagonal measure to fibre level sets is false, breaking the first inclusion of Theorem 1.6.","tokens_in":18077,"feed_emoji":"🔁","tokens_out":20370,"duration_ms":185140,"temperature":0.7,"pith_summary":"This paper sets out to unify three local ways of measuring complexity in a topological dynamical system: sequence entropy tuples, mean-sensitive tuples, and sensitive-in-the-mean tuples. It proves that for an ergodic invariant measure on a system acted on by a countable infinite amenable group, all three notions coincide along any tempered Følner sequence. It also proves the half of the story that needs no amenability: every measure-theoretic sequence entropy tuple is an independence (IT) tuple. A reader should care because this turns sequence entropy, which is defined by a supremum over all time sequences, into a single orbit-averaging condition that can be checked along one fixed averaging schedule.","feed_headline":"Sequence entropy equals mean sensitivity for ergodic actions","feed_subtitle":"Proof covers all K-tuples and non-abelian amenable groups, unifying three local complexity notions.","key_machinery":"The object that carries the argument is the diagonal measure $\\lambda^X_K=\\int_X \\mu_x\\times\\cdots\\times\\mu_x\\,d\\mu(x)$, built from the disintegration of $\\mu$ over its Kronecker factor (the largest factor with discrete spectrum); its support, minus the diagonal, is exactly the set of sequence-entropy tuples. The proof of Theorem 1.6 represents the Kronecker factor as a rotation on a compact homogeneous space $Z/H$, decomposes each product measure $\\mu^{(K)}$ into ergodic components $\\lambda^X_z$ indexed by $z\\in Z^K$, and uses a level-set lemma to turn a positive value of $\\lambda^X_K$ on a product of neighborhoods into a large set of fibers where every coordinate has uniformly positive mass. The converse direction uses the fact that a zero-sequence-entropy partition lies inside the Kronecker algebra, so its indicator functions are uniformly almost periodic along the tempered Følner sequence; an averaging contradiction then forces the tuple to be mean sensitive.","core_discovery":"The central claim, Theorem 1.6, is that for a topological dynamical system $(X,G)$ with $G$ amenable and $\\mu$ ergodic, the set $\\mathrm{SE}^\\mu_K(X,G)$ of $\\mu$-sequence-entropy $K$-tuples, the set $\\mathrm{MS}^\\mu_K(X,F)$ of $\\mu$-mean-sensitive $K$-tuples along a tempered Følner sequence $F$, and the set $\\mathrm{SM}^\\mu_K(X,F)$ of $\\mu$-sensitive-in-the-mean $K$-tuples along $F$ are the same for every $K\\ge2$. The supporting result Theorem 1.1 shows that $\\mathrm{SE}^\\mu_K(X,G)\\subset \\mathrm{IT}_K(X,G)$, where IT tuples are those whose neighborhoods admit infinite independence sets, so that every sequence-entropy tuple is an IT tuple; together with the known inclusion $\\mathrm{IT}_K\\subset \\mathrm{SE}_K$ this closes the chain of local complexity notions. Theorem 1.3 and Corollary 1.5 convert these local results into a global bound: if no essential IT $K$-tuples exist, then $h^*_\\mu(G)\\le \\log(K-1)$ for every invariant measure $\\mu$.","pith_inferences":["Because the inclusion SE⊂IT does not use amenability, the paper suggests that the independence character of sequence-entropy tuples is a non-amenable phenomenon; a natural test is whether the converse half of Theorem 1.6 can be formulated with invariant means in place of Følner sequences.","The fibre-shifting construction indicates an algorithmic way to certify high sequence entropy: sample fibers of the Kronecker factor and look for tuples whose neighborhoods have uniformly positive mass on a large level set; this could be implemented numerically for low-dimensional systems.","For non-ergodic measures the coincidence is known to fail, so the paper's use of ergodicity is not merely technical; extending the result would require an additional assumption controlling how Kronecker fibers vary across the ergodic decomposition.","The equality of the three tuple sets gives a new local analogue of the global statement that positive sequence entropy is equivalent to mean sensitivity, suggesting that further local-global bridge theorems for amenable group actions may hold for other complexity notions such as weak mixing or Devaney chaos."],"forward_implications":["For ergodic measures on amenable groups, sequence-entropy tuples are computable by a fixed Følner averaging: a tuple is such a tuple exactly when every positive-measure set contains points that jointly visit the tuple's neighborhoods with positive limsup frequency.","Every measure-theoretic sequence entropy tuple is an IT tuple, so the measure-theoretic and combinatorial notions of independence are connected: $\\bigcup_\\mu \\mathrm{SE}^\\mu_K \\subset \\mathrm{IT}_K \\subset \\mathrm{SE}_K$.","Systems without essential IT K-tuples have maximal sequence entropy at most $\\log(K-1)$ for every invariant measure, giving a multi-tuple version of Huang's tameness bound.","If an ergodic system has $h^*_\\mu(G)>\\log(K-1)$, it must contain essential mean-sensitive K-tuples, so high sequence entropy manifests as a local sensitivity phenomenon.","The mean-sensitive and sensitive-in-the-mean notions, previously known to be equivalent for pairs in abelian systems, now coincide with sequence entropy tuples for every K in all amenable group actions."],"supporting_citations":[{"why":"Supplies the definition of IT tuples and the result that IT tuples are topological sequence entropy tuples, against which the inclusion SE⊂IT is measured.","marker":"[19]"},{"why":"Supplies tempered Følner sequences and the pointwise ergodic theorem used to pass from ergodic measures on the product to individual orbit-average frequencies.","marker":"[26]"},{"why":"Gives the compact homogeneous-space model for ergodic discrete-spectrum systems that underlies the decomposition into the measures $\\lambda^X_z$.","marker":"[30]"},{"why":"Provides the uniform almost-periodicity lemma for Kronecker-algebra indicators that drives the contradiction in the second inclusion.","marker":"[31]"},{"why":"Supplies the mean-sensitive tuple framework and the proof strategy used in the sensitive-in-the-mean to sequence-entropy direction.","marker":"[25]"},{"why":"Establishes the formula $h^*_\\mu(G,\\alpha)=H_\\mu(\\alpha|K_\\mu(G))$ for general countable infinite discrete groups, connecting zero sequence entropy to the Kronecker algebra.","marker":"[27]"},{"why":"Gives the combinatorial proof of the same sequence-entropy formula in measurable dynamics, an alternative load-bearing route used by the authors.","marker":"[20]"},{"why":"Provides the earlier Z-action version of Proposition 2.7 identifying sequence-entropy tuples with the support of the diagonal measure.","marker":"[16]"},{"why":"Supplies the Kronecker-algebra product identity for product systems used to decompose $\\mu^{(K)}$ into ergodic components.","marker":"[7]"}],"fun_headline_variants":["Ergodic actions: sequence entropy, mean sensitivity coincide","Three local complexity notions unify for ergodic amenable actions","Sequence entropy tuples equal mean sensitivity tuples for ergodic measures","IT tuples, sequence entropy, mean sensitivity: one set for ergodic actions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The equality depends on two structural inputs: the almost-periodic factor of an ergodic system being a compact-group rotation (so fibre masses can be shifted and averaged), and its indicator functions being uniformly almost periodic along the averaging sequence; if either input fails for a concrete system, one of the two inclusions breaks.","fun_headline_variants_meta":{"raw":{"variants":["Ergodic actions: sequence entropy, mean sensitivity coincide","Three local complexity notions unify for ergodic amenable actions","Sequence entropy tuples equal mean sensitivity tuples for ergodic measures","IT tuples, sequence entropy, mean sensitivity: one set for ergodic actions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000686,"raw_usage":{"total_tokens":3078,"prompt_tokens":881,"completion_tokens":2197,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":497,"completion_tokens_details":{"reasoning_tokens":2126}},"tokens_in":497,"tokens_out":2197,"duration_ms":12752,"temperature":1.0,"reasoning_tokens":2126,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:30:22.774358+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for an ergodic rotation on a compact abelian group (say the circle) and two small arcs, the diagonal measure of the product of the arcs and the Haar measure of the set of rotations for which both fibres have mass greater than $b$; the proof predicts the latter exceeds $b$ whenever the former exceeds $2b$. A concrete example violating this inequality would show that the step from diagonal measure to fibre level sets is false, breaking the first inclusion of Theorem 1.6.","supporting_citations":[{"cited_title":"Lindenstrauss, Pointwise theorems for amenable groups","cited_arxiv_id":null,"evidence_quote":"Supplies tempered Følner sequences and the pointwise ergodic theorem used to pass from ergodic measures on the product to individual orbit-average frequencies."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the compact homogeneous-space model for ergodic discrete-spectrum systems that underlies the decomposition into the measures $\\lambda^X_z$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the uniform almost-periodicity lemma for Kronecker-algebra indicators that drives the contradiction in the second inclusion."},{"cited_title":"Liu and X","cited_arxiv_id":null,"evidence_quote":"Establishes the formula $h^*_\\mu(G,\\alpha)=H_\\mu(\\alpha|K_\\mu(G))$ for general countable infinite discrete groups, connecting zero sequence entropy to the Kronecker algebra."},{"cited_title":"Kerr and H","cited_arxiv_id":null,"evidence_quote":"Gives the combinatorial proof of the same sequence-entropy formula in measurable dynamics, an alternative load-bearing route used by the authors."},{"cited_title":"Huang, A","cited_arxiv_id":null,"evidence_quote":"Provides the earlier Z-action version of Proposition 2.7 identifying sequence-entropy tuples with the support of the diagonal measure."},{"cited_title":"Glasner, Ergodic theory via joinings, Mathematical Surveys a nd Monographs, Vol","cited_arxiv_id":null,"evidence_quote":"Supplies the Kronecker-algebra product identity for product systems used to decompose $\\mu^{(K)}$ into ergodic components."}],"review_version":1}