Coexistence of mixing and rigid behaviors in ergodic theory

Rigoberto Zelada

REVIEW 2 major objections 5 minor 37 references

Coexistence of mixing and rigid behaviors in ergodic theory

T0 review · 2 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2504.17555 v2 pith:QC7V2EXU submitted 2025-04-24 math.DS

Rigoberto Zelada This is my paper · ORCID

classification math.DS MSC 37A0537A2537A3005D1011B30

keywords rigiditygroupadequatesequencesgenerictransformationsmixingandIP-ergodictheorymultiplerecurrencespectralErgodic-Ramsey

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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 $\deg(2p-q)<\deg(p)$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The 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.

That 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.

Other 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.

Who 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.

Referee Report

2 major / 5 minor

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).

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 (2)

[§5.4, Theorem 1.5] 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.
[Sections 3 and 4.2] 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.

minor comments (5)

[Global] 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.
[§5.4] 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.
[§5.3.1] 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.
[Abstract and §1] 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.
[References] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the central characterization is derived from first principles and self-citations are to independently published tools.

full rationale

Theorem 1.8, the main load-bearing result, is proved in Sections 2-3 rather than imported: the hard direction (ii)=>(iii) is reduced to Lemma 2.1, whose construction of sigma and (n_k) is carried out in the paper using only Weyl's uniform-distribution lemma (Lemma 2.2, cited to [33] as 'Cf. Theorem 21 in [33]') plus elementary Diophantine estimates. The converse (iv)=>(i) constructs the unitary U from the spectral measure sigma, and (iii)=>(iv) is trivial; no direction defines A(phi) in terms of H(U,(n_k)) or vice versa beyond the immediate (i)=>(ii) inclusion. The applications (Theorems 4.1, 1.5, Corollaries 1.9, 1.10, 1.14, 1.15) use the author's published [34] for Theorem 4.4 and Theorem 5.3, but those are independent peer-reviewed results whose hypotheses are verified inside the present paper (e.g., Corollary 5.1 establishes condition (c) before invoking [34, Theorem 1.3]); this is legitimate external support, not a self-citation chain forcing the conclusion. The abstract's no-orthogonal-vector criterion is a corollary of Theorem 4.1, not an input. The only flagged issue is in Section 5.4, where '(3) =⇒ (1) follows from the proof of (5.6) (see Subsubsection 5.3.1) and (c) =⇒ (a) in Corollary 5.1' is abbreviated and arguably incomplete as an argument; however, incompleteness is a correctness gap, not circularity, because the cited proof does not assume the conclusion (1) or define it into the hypothesis (3). No fitted parameter is renamed as a prediction, no known result is merely renamed, and no ansatz is smuggled in via self-citation.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim relies on standard tools from ergodic theory, harmonic analysis, and uniform distribution. No empirical free parameters are fitted; all numerical choices, such as M = product of b_j, α_k^(j), and n_k, are constructed in the proofs. The only new notion introduced is the definition of 'rigidity group', which is a mathematical definition, not an unexplained empirical entity.

assumptions (7)

standard math Weyl's uniform distribution theorem (Lemma 2.2, from [34] Theorem 21, originally Weyl [33])
Used in Lemma 2.1 Step 1 to choose irrationals α with uniform distribution properties (σ.5).
standard math Bochner's theorem for positive definite functions (Corollary 3.2, see [30] Section 1.4)
Used to deduce unitary characterizations (iii') and (iv') from the spectral measure statement (iii)-(iv).
standard math Halmos conjugacy lemma (Lemma 5.5, [24] page 77)
Used in Lemma 5.4 to conjugate transformations and obtain a dense approximating family.
standard math Rohlin's lemma (proof of Lemma 5.4, [24] page 71)
Used to prove adequacy of sequences from the dense G_δ property by producing a set A disjoint from T^{-b}A.
domain assumption Gaussian systems construction (Subsection 4.2, [14] Chapters 8, [26] Appendix C)
Used in Theorem 4.1 and Corollary 5.1 to convert unitary weak limits into mixing/rigid behavior of measure-preserving transformations.
standard math Hindman's theorem (Section 6, [25]) and IP-limit compactness
Used in Lemma 6.1 and Theorem 6.2 to pass from regular limits to IP-limits.
standard math Lemma 4.3 (Lemma 3.15 in [13]) on separation by finite-index subgroups
Used in Lemma 4.2 to extend a subgroup G with A(φ) subset of G to a finite-index subgroup excluding specified elements.

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Cite this review

Pith. "Pith review of Coexistence of mixing and rigid behaviors in ergodic theory." pith.science (2026). https://pith.science/paper/QC7V2EXU

@misc{pith2026250417555,
  author       = {Pith},
  title        = {Pith review of: Coexistence of mixing and rigid behaviors in ergodic theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QC7V2EXU}},
  note         = {Machine review of arXiv:2504.17555}
}

abstract

In this paper we introduce and explore the notion of rigidity group, associated with a collection of finitely many sequences, and show that this concept has many, somewhat surprising characterizations of algebraic, spectral, and unitary nature. Furthermore, we demonstrate that these characterizations can be employed to obtain various results in the theory of generic Lebesgue-preserving automorphisms of $[0,1]$, IP-ergodic theory, multiple recurrence, additive combinatorics, and spectral theory. As a consequence of one of our results we show that given $(b_1,...b_\ell)\in\mathbb N^\ell$, there is no orthogonal vector $(a_1,\dots,a_\ell)\in\mathbb Z^\ell$ with some $|a_j|=1$ if and only if there is an increasing sequence of natural numbers $(n_k)_{k\in\mathbb N}$ with the property that for each $F\subseteq \{1,...,\ell\}$ there is a $\mu$-preserving transformation $T_F:[0,1]\rightarrow[0,1]$ ($\mu$ denotes the Lebesgue measure) such that for any measurable $A,B\subseteq [0,1]$, $$\lim_{k\rightarrow\infty}\mu(A\cap T_F^{-b_jn_k}B)=\begin{cases} \mu(A\cap B),\,\text{ if }j\in F,\\ \mu(A)\mu(B),\,\text{ if }j\not\in F. \end{cases}$$ We remark that this result has a natural extension to a wide class of families of sequences.

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With an appendix by Imre Ruzsa

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [34]

R. Zelada. Mixing and rigidity along asymptotically li nearly independent sequences. Ergodic Theory and Dynamical Systems , 43(10):3506–3537, 2023

work page 2023

[2] [35]

R. Zelada. Failure of Khintchine-type results along th e polynomial image of IP 0 sets. Discrete and Continuous Dynamical Systems , 44(5):1475–1494, 2024

work page 2024

[3] [8]

Bergelson and R

V. Bergelson and R. Zelada. Ramsey theoretical reﬁnement s of the polynomial Szemerédi theorem. In preparation

work page

[4] [30]

W. Rudin. Fourier analysis on groups . John Wiley and sons, 1962

work page 1962

[5] [1]

J. A vigad. Inverting the Furstenberg correspondence. Discrete and Continuous Dynamical Systems, 32(10):3421–3431, 2012

work page 2012

[6] [2]

F. A. Behrend. On sets of integers which contain no three te rms in arithmetical progression. Proceedings of the National Academy of Sciences of the Unite d States of America , 32:331–332, 1946

work page 1946

[7] [3]

Bergelson

V. Bergelson. Weakly mixing PET. Ergodic Theory and Dynamical Systems , 7(3):337–349, 1987

work page 1987

[8] [4]

Bergelson

V. Bergelson. Ergodic Ramsey theory—An Update. In Ergodic Theory of Zd-Actions, volume 228, pages 1–61, Warwick, 1993-1994, 1996. London Mathemat ical Society Lecture Note Series, Cambridge University Press

work page 1993