{"id":"715d5b02-d507-43f4-9ea7-7b20453af1c5","arxiv_id":"2504.17555","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A characterization theorem showing that for adequate integer sequences, a subgroup condition exactly determines when a unitary operator or measure-preserving transformation can be rigid along some given subsequences and mixing along others.","lead":"This paper introduces a notion called the rigidity group of a finite list of integer sequences and proves that it has several equivalent algebraic, spectral, and unitary descriptions. The main results give a clean test for when a single volume-preserving transformation can mix along some prescribed time sequences while staying rigid along others, with applications to multiple recurrence.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.5's reverse direction (3)=>(1) is not actually proved: the text delegates it to the proof of (5.6), which assumes the standing condition (5.3) that is exactly part of the conclusion. This leaves the full equivalence incomplete.","rationale":"The reader's conditional verdict is justified. I found no fatal flaw in the central equivalence Theorem 1.8: the construction in Lemma 2.1 is elaborate but internally coherent, and the reduction in Theorem 3.1 through asymptotically independent subfamilies and the subgroup G̃ is sound at the level of the written algebra. The most exposed point is the treatment of Theorem 1.5, which the abstract presents as a headline consequence. The reverse direction (3)=>(1) is dispatched with a reference to a proof that begins with the very assumption it needs to establish, and that proof additionally relies on the dense-Gδ hypothesis (2). This is not merely an omitted routine detail: it concerns whether the stated equivalence is actually established for all families satisfying (3). The reader already flagged the standing assumption (1.5) as a restriction; my concern is that even within the theorem's own proof, the reverse direction is incomplete. This supports the same CONDITIONAL verdict but shifts the focus from restrictiveness to an unproven implication. Completing the proof would require a direct argument deriving (5.3) from (3), or a counterexample showing the need to retain (1.5) in the statement.","tokens_in":33741,"tokens_out":31177,"duration_ms":312513,"concrete_test":"Independently derive (1.5) from condition (3) of Theorem 1.5, using only the existence of the common increasing sequence (n_k) and weakly mixing T_λ with the prescribed limits; do not import the standing assumption from §5.3.1. In particular, check whether the argument can force lim_n Σ a_j φ_j(n) to exist for all a when φ is not assumed adequate. If the only derivation goes through (5.3) or the dense-Gδ condition (2), the proof of (3)=>(1) is circular and must be completed or the theorem qualified.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section 5.4 states that (3) =⇒ (1) in Theorem 1.5 'follows from the proof of (5.6) (see Subsubsection 5.3.1) and (c) =⇒ (a) in Corollary 5.1.' The referenced proof in §5.3.1, however, is run under the standing assumption (5.3): for every a_1,...,a_ℓ ∈ Z, lim_n Σ a_j φ_j(n) exists as an integer, ∞, or −∞. That is precisely a conclusion that would need to be derived from (3), but no derivation is supplied. The same subsubsection also uses the dense-Gδ condition (2) to cite Lemma 5.2 by way of Corollary 5.1; condition (3) only gives the existence of one T_λ in each G_λ, not the dense Gδ membership. The no-±1 part of (1) can be recovered from (3) via Corollary 5.1(c)=>(a), since the T_λ for 0/1-valued λ are weakly mixing and hence aperiodic, but the adequacy of φ_1,...,φ_ℓ (the remaining content of (1)) is never established. Thus the reverse implication of Theorem 1.5 is incomplete as written: the stated equivalence may be true, but the manuscript does not prove it.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines the notion of a (φ_1,...,φ_ℓ)-rigidity group for adequate sequences φ_j:N→Z and a subgroup G of Z^ℓ. Its main technical result, Theorem 1.8, characterizes rigidity groups algebraically by the inclusion A(φ_1,...,φ_ℓ)⊆G, dynamically by the existence of a measure σ on T whose weak limits along a sequence n_k equal Haar measure on the annihilator of G, and unitarily by weak limits of powers U^{Σ a_j φ_j(n_k)}. The proof is carried out in Sections 2-3, with Lemma 2.1 giving a detailed construction for asymptotically linearly independent sequences and Theorem 3.1 providing a reduction to that case. The paper then applies this dictionary to prove a generalized version of a theorem of Bergelson-Kasjan-Lemańczyk (Theorem 4.1), a characterization of dense Gδ sets of transformations with prescribed rigid/mixing behavior (Theorem 1.5), and several IP*-recurrence results (Corollaries 1.10, 1.14, 1.15).","tokens_in":33995,"tokens_out":16620,"duration_ms":151520,"significance":"Theorem 1.8 is a useful and nontrivial transfer principle: it converts an algebraic inclusion into the existence of unitary operators and measures with prescribed weak limits along a common sequence, and the construction of the measure in Lemma 2.1 is presented in detail. The paper also gives concrete, falsifiable consequences for generic transformations and for IP*-recurrence, and it is candid about provenance, including the announcement of joint work [8]. If the applications are fully validated, the paper will be a substantial contribution to ergodic-ramsey theory and the study of generic transformations. However, the advertised full equivalence in Theorem 1.5 is not proved as written, and some auxiliary results, namely Corollary 3.2 and Theorem 4.4, are used without full proofs; these defects are local but must be repaired before the paper can be accepted.","major_comments":[{"comment":"The proof of (3) =⇒ (1) in Theorem 1.5 is incomplete as written. The text states that the implication follows from the proof of the adequacy claim in Subsubsection 5.3.1 and from (c) =⇒ (a) in Corollary 5.1. However, the adequacy proof in Subsubsection 5.3.1 is run inside Lemma 5.4 under the dense-Gδ hypothesis (2), whereas condition (3) only yields one weakly mixing transformation T_λ for each λ. In particular, the argument there invokes the full family of dense Gδ sets G_λ for subsequences in order to apply Lemma 5.2; condition (3) does not provide those dense Gδ sets. The direct derivation of adequacy from (3) together with the standing assumption (1.5) is not supplied, and it is not a purely formal consequence of the displayed argument. Additionally, the proof of (c) =⇒ (a) in Corollary 5.1 is only sketched by reference to a similar argument, even though this implication is used in the same reverse direction. Please provide a complete proof of (3) =⇒ (1), or replace Theorem 1.5 by the weaker statement that is actually proved.","section":"§5.4, Theorem 1.5"},{"comment":"The manuscript states Corollary 3.2 with 'We omit the proofs' and Theorem 4.4 with 'We omit the proof.' These results are load-bearing for the applications: Lemma 4.2(a) =⇒ (b) explicitly invokes Corollary 3.2, and Theorem 4.4 is the bridge from unitary weak limits to the mixing and rigidity behavior of Gaussian systems used in Theorem 4.1 and Corollary 5.1. As written, a reader cannot verify these steps without consulting [34] and reconstructing the Bochner/positive-definiteness argument. Please include full proofs or state the exact results from [34] and [30] that are being used, and indicate which parts of the claimed equivalence depend on them.","section":"Sections 3 and 4.2"}],"minor_comments":[{"comment":"The manuscript contains many typographical errors and OCR artifacts: 'defined by' spacing, 'defined' for 'defined', 'striclty', 'followng', 'defiend', and garbled symbols such as '/BD E(x)' for 1_E and '/BD T' for 1_T. A careful proofreading pass is needed before publication.","section":"Global"},{"comment":"The reference to 'the proof of (5.6)' in the proof of Theorem 1.5 is dangling, because no displayed equation (5.6) appears in the manuscript. Please refer to the specific claim in Subsubsection 5.3.1, for example the assertion that the sequences in (5.6) are adequate, or renumber the displayed statements accordingly.","section":"§5.4"},{"comment":"In the proof of Corollary 5.1(b) =⇒ (c), the argument adds a sequence φ_{ℓ+1} with a strong domination property but does not justify its existence; this is easy to supply, for example φ_{ℓ+1}(n)=n(1+Σ_j |φ_j(n)|), but the construction should be stated explicitly.","section":"§5.3.1"},{"comment":"The abstract's display uses the notation T_F^{-b_j n_k} and the body uses /BD_E(x) and /BD_T; these should be typeset as T_F^{-b_j n_k}, 1_E(x), and 1_T so that the intended mathematical content is readable.","section":"Abstract and §1"},{"comment":"Reference [8] is listed as 'In preparation' and is used both for the provenance of Corollary 1.10 and for a strengthened result in Remark 1.16; this is acceptable, but the dependence on unpublished work should be explicitly flagged in the main text.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper relies heavily on the author's own earlier publications [34] and [35] as well as the in-preparation joint work [8] for results used inside the main proofs. This is not by itself a problem, but the editor may want to check that the overlap with [34] and [13] is sufficiently delineated. The reverse direction of Theorem 1.5 should be fixed before publication; the remaining omissions, while local, affect the self-containedness of the applications."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe real substance here is Theorem 1.8: for adequate sequences, being a rigidity group is equivalent to an algebraic inclusion A(φ)⊂G and to existence of a measure with prescribed Fourier limits. This is a genuine extension of the Bergelson–Kasjan–Lemańczyk framework, and the proof is mostly detailed, especially the Diophantine construction of the measure in Lemma 2.1. The applications to generic transformations and to IP-recurrence are plausible and give the paper range.\n\nThat said, the stress-test note is correct: Theorem 1.5's reverse direction (3)=>(1) is not proved. Section 5.4 delegates to the proof of (5.6) in §5.3.1, but that proof runs under the standing assumption (5.3) that every integer combination of the φ_j has a limit in Z∪{±∞}. That assumption is essentially part of what (1) asserts, and condition (3) does not give it. Moreover, condition (3) only supplies one weakly mixing T_λ per λ, not the dense Gδ membership needed to invoke Lemma 5.2 or Corollary 5.1. The no-±1 part of (1) can likely be recovered from (3) via aperiodicity of weakly mixing transformations, but the adequacy of the φ_j is not derived. So the equivalence as stated is incomplete.\n\nOther soft spots are minor by comparison: Corollary 3.2 and Theorem 4.4 have explicitly omitted proofs (delegated to Bochner's theorem and to [34]), and Lemma 6.1 is also stated without proof. These are supporting rather than load-bearing for Theorem 1.8, but they should be filled in or given precise references before the paper is treated as verified. The self-citations to [34] and [35] are frequent but are used as tools, not as assumptions equivalent to the conclusions; I do not see a circularity problem.\n\nWho should read this: ergodic theorists working on rigidity, mixing, and IP-recurrence will get real value from Theorem 1.8 and its applications. The paper deserves a serious referee, and the right outcome is probably major revision: keep Theorem 1.8, fix or restrict Theorem 1.5, and complete the omitted supporting proofs. I would send it to an expert rather than desk-reject, but I would not accept it as is.","headline":"Theorem 1.8 gives a genuinely useful algebraic/spectral/unitary characterization of rigidity groups, but the proof of Theorem 1.5 is incomplete in the (3)=>(1) direction, exactly as the stress-test note says.","tokens_in":34571,"tokens_out":3764,"would_cite":true,"duration_ms":38011,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37A05","37A25","37A30","05D10","11B30"],"pacs":[],"model":"deepseek-v4-flash","headline":"A subgroup $G$ of $\\mathbb{Z}^\\ell$ is a rigidity group exactly when it contains $A(\\varphi_1,\\ldots,\\varphi_\\ell)$, the set of coefficient vectors whose combination tends to zero; that algebra also dictates the unitary and…","keywords":["rigidity group","adequate sequences","generic transformations","mixing and rigidity","IP-ergodic theory","multiple recurrence","spectral theory","Ergodic-Ramsey theory"],"falsifier":"To test the main criterion, fix $\\varphi_1(n)=n$ and $\\varphi_2(n)=2n$. The vector $(2,-1)$ lies in $A(\\varphi_1,\\varphi_2)$, so Theorem 1.5 predicts that no measure-preserving transformation $T$ and increasing times $n_k$ can satisfy $\\lim_k T^{n_k}\\to\\mathrm{Id}$ and $\\lim_k T^{2n_k}\\to 0$ in the weak operator topology on $L^2_0(\\mu)$. Constructing such a transformation, or any system with this mixed limiting behavior, would refute the necessity of the $\\pm1$ orthogonal-vector condition; checking the impossibility directly for this pair would confirm the obstruction.","tokens_in":33474,"feed_emoji":"🔄","tokens_out":11182,"duration_ms":103637,"temperature":0.7,"pith_summary":"This paper is trying to establish that one algebraic inclusion governs whether mixing and rigidity can coexist along a prescribed family of sequences. For sequences $\\varphi_1,\\ldots,\\varphi_\\ell$, the paper defines $A(\\varphi)$ as the subgroup of coefficient vectors $(a_1,\\ldots,a_\\ell)$ for which $\\sum_j a_j\\varphi_j(n)$ tends to zero, and calls a subgroup $G$ of $\\mathbb{Z}^\\ell$ a rigidity group when some unitary operator, along one increasing sequence of times, converges to the identity precisely on $G$. The central result is that $G$ is a rigidity group exactly when $A(\\varphi)\\subseteq G$, and this is equivalent to a measure-theoretic identity: integrals against a measure $\\sigma$ on the circle, evaluated along the $\\varphi$-weighted times, converge to integrals against the Haar measure on the annihilator of $G$. A corollary gives a clean answer to when every interpolation between mixing and rigidity, parameterized by $\\lambda\\in[0,1]^\\ell$, is realized by generic measure-preserving transformations. The paper then uses these equivalences to build multiple-recurrence examples where return sets fail to be IP*, sharpening known syndetic and almost-IP* results.","feed_headline":"Subgroup inclusion decides when sequences mix or stay rigid","feed_subtitle":"Inclusion of A(φ) in G is exactly what turns algebra into unitary operators with prescribed limits.","key_machinery":"The load-bearing pair of objects is $H(U,(n_k))$, the subgroup of coefficient vectors along which powers $U^{\\sum_j a_j\\varphi_j(n_k)}$ converge strongly to the identity, and $A(\\varphi_1,\\ldots,\\varphi_\\ell)$, the subgroup of vectors for which the weighted combination of sequences converges to zero. The main equivalence is the bridge $A(\\varphi)\\subseteq G$, reduced to the annihilator of $G$ through $\\lambda_G$. The measure $\\sigma$ is constructed by a continuous map from a product of finite $k!$-ary grids into the circle; the uniform-distribution properties of strongly asymptotically independent sequences ensure that the $\\varphi_j(n_k)$-pushforwards sample the annihilator of $G$. Bochner's theorem then converts the measure identity into the spectral characterization of unitary operators.","core_discovery":"The central claim, stated as Theorem 1.8, is that for adequate sequences $\\varphi_1,\\ldots,\\varphi_\\ell:\\mathbb{N}\\to\\mathbb{Z}$, the following are equivalent for a subgroup $G$ of $\\mathbb{Z}^\\ell$: $G$ contains the subgroup $A(\\varphi)$ of coefficient vectors whose weighted combination tends to zero; $G$ is the rigidity group of some unitary operator, meaning $G=H(U,(n_k))$ for an increasing sequence $(n_k)$; and there is a Borel probability measure $\\sigma$ on the circle and times $(n_k)$ such that for every continuous $f$ and measurable $E$, $\\lim_k \\int_T \\mathbf{1}_E(x) f(\\varphi_1(n_k)x,\\ldots,\\varphi_\\ell(n_k)x)\\,d\\sigma = \\sigma(E)\\int_{T^\\ell} f\\,d\\lambda_G$, with $\\lambda_G$ the Haar measure on the annihilator of $G$. The proof first treats asymptotically linearly independent sequences, constructing $\\sigma$ from a product of $k!$-adic grid measures whose Diophantine behavior reproduces the annihilator of $G$, and then reduces the general adequate case to that situation through an intermediate subgroup reduction. From this core, the paper derives the orthogonal-vector criterion for generic transformations: absence of coefficient vectors with a $\\pm1$ entry in $A(\\varphi)$ is equivalent to the existence of a single sequence of times realizing every $\\lambda$-weight between mixing and rigidity for weakly mixing transformations, with the realizing transformations forming dense $G_\\delta$ sets.","pith_inferences":["One could test whether the rigidity-group criterion extends to non-adequate sequences by replacing ordinary limits with IP-limits or other generalized convergence; the paper's factorial-based IP construction suggests a template for such an extension.","The irreducibility assumption on sequences could be probed by trying to construct a dense $G_\\delta$ set $G_\\lambda$ for sequences whose integer combinations have bounded oscillatory parts; the paper's remark that every sequence has an irreducible subsequence indicates such a construction, if it exists, would have to use generalized convergence rather than ordinary limits.","The IP-ergodic machinery should transfer to families such as sequences of the form $\\lfloor n\\alpha_j\\rfloor$ with rationally independent $\\alpha_j$, replacing the $k!$-adic construction with a Diophantine one; the polynomial examples in the paper are the first test cases."],"forward_implications":["For adequate sequences, the subgroup $A(\\varphi)$ is the only obstruction: any subgroup of $\\mathbb{Z}^\\ell$ containing it can be realized as a rigidity group, and any realized rigidity group must contain it.","There is a single increasing sequence of times along which every interpolation between mixing and rigidity, parameterized by $\\lambda\\in[0,1]^\\ell$, is realized by some weakly mixing transformation; for each $\\lambda$ the set of such transformations is dense $G_\\delta$ in $\\mathrm{Aut}([0,1],\\mu)$.","Linearly independent integer polynomials with zero constant term produce measure-preserving systems whose multiple-recurrence return sets fail to be IP*, even though the same sets are syndetic and almost IP*, showing a sharp boundary for ergodic-Ramsey phenomena.","Infinite-index rigidity groups still carry the full equivalent characterizations, yielding systems where correlated returns fail IP* for polynomials $p,q$ with equal degree and $\\deg(2p-q)<\\deg(p)$."],"supporting_citations":[{"why":"defines the prior N-rigidity group and global rigidity notions, and supplies Theorem E and Lemma 3.15, the structural starting point the paper's rigidity group extends.","marker":"[13]"},{"why":"supplies the dense-$G_\\delta$ theorem for prescribed mixing/rigidity limits and the uniform distribution lemma used to construct the measure $\\sigma$ on the circle.","marker":"[34]"},{"why":"provides the uniform-distribution criterion that Lemma 2.2 uses to select the Diophantine coefficients in the $k!$-adic grid construction.","marker":"[33]"},{"why":"supplies the annihilator duality for subgroups of $\\mathbb{Z}^\\ell$ and Bochner's theorem, which convert the measure identity into spectral and unitary characterizations.","marker":"[30]"},{"why":"provides the conjugacy lemma and the completeness of the weak topology on $\\mathrm{Aut}([0,1],\\mu)$, used to produce one universal sequence realizing all $\\lambda$-parameters.","marker":"[24]"},{"why":"provides the multiple-recurrence theorem that serves as template and ingredient for the IP*-failure examples in Section 6.","marker":"[10]"}],"fun_headline_variants":["Subgroup condition unifies mixing and rigidity in ergodic theory","When sequences mix or stay rigid: a subgroup criterion","Rigidity groups: algebra decides limits for generic transformations","No unit orthogonal vector enables mixing-rigidity blends","One time sequence realizes every mix-rigidity blend"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that every integer combination of the sequences has a definite limiting value, either an integer or plus or minus infinity; sequences whose combinations oscillate inside a bounded range are excluded, and the main equivalences and the $\\pm1$ test are proved only under this irreducibility condition.","fun_headline_variants_meta":{"raw":{"variants":["Subgroup condition unifies mixing and rigidity in ergodic theory","When sequences mix or stay rigid: a subgroup criterion","Rigidity groups: algebra decides limits for generic transformations","No unit orthogonal vector enables mixing-rigidity blends","One time sequence realizes every mix-rigidity blend"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00064,"raw_usage":{"total_tokens":3054,"prompt_tokens":1163,"completion_tokens":1891,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":779,"completion_tokens_details":{"reasoning_tokens":1812}},"tokens_in":779,"tokens_out":1891,"duration_ms":14499,"temperature":1.0,"reasoning_tokens":1812,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:37:15.001497+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"To test the main criterion, fix $\\varphi_1(n)=n$ and $\\varphi_2(n)=2n$. The vector $(2,-1)$ lies in $A(\\varphi_1,\\varphi_2)$, so Theorem 1.5 predicts that no measure-preserving transformation $T$ and increasing times $n_k$ can satisfy $\\lim_k T^{n_k}\\to\\mathrm{Id}$ and $\\lim_k T^{2n_k}\\to 0$ in the weak operator topology on $L^2_0(\\mu)$. Constructing such a transformation, or any system with this mixed limiting behavior, would refute the necessity of the $\\pm1$ orthogonal-vector condition; checking the impossibility directly for this pair would confirm the obstruction.","supporting_citations":[{"cited_title":"Bergelson, S","cited_arxiv_id":null,"evidence_quote":"defines the prior N-rigidity group and global rigidity notions, and supplies Theorem E and Lemma 3.15, the structural starting point the paper's rigidity group extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the dense-$G_\\delta$ theorem for prescribed mixing/rigidity limits and the uniform distribution lemma used to construct the measure $\\sigma$ on the circle."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the uniform-distribution criterion that Lemma 2.2 uses to select the Diophantine coefficients in the $k!$-adic grid construction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the annihilator duality for subgroups of $\\mathbb{Z}^\\ell$ and Bochner's theorem, which convert the measure identity into spectral and unitary characterizations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the conjugacy lemma and the completeness of the weak topology on $\\mathrm{Aut}([0,1],\\mu)$, used to produce one universal sequence realizing all $\\lambda$-parameters."},{"cited_title":"Bergelson, B","cited_arxiv_id":null,"evidence_quote":"provides the multiple-recurrence theorem that serves as template and ingredient for the IP*-failure examples in Section 6."}],"review_version":1}