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Resolving the joint ergodicity problem for Hardy sequences

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Pith's one-line read This paper proves that Hardy sequences of polynomial growth are jointly ergodic whenever they satisfy the difference and product ergodicity conditions, and it constructs a counterexample showing the converse fails in general.

desk verdict Important theorem and genuinely new tools, but the proof of the relative concatenation theorem contains a load-bearing approximation claim that is false as stated; this needs serious fixing before the main result can be trusted. read the letter →

arxiv 2506.20459 v2 pith:RPP4FMYK submitted 2025-06-25 math.DS

Sebastián Donoso , Andreas Koutsogiannis , Borys Kuca , Wenbo Sun , Konstantinos Tsinas This is my paper · ORCID
classification math.DS MSC 37A4411B3028D05
keywords jointergodicityHardysequencesfieldsmultipleergodicaveragesHost-Kraseminormsboxseminormsmoothingconcatenationtheorems
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

Joint ergodicity — the convergence of multiple ergodic averages along a tuple of sequences to the product of the integrals — is a central notion in ergodic theory. This paper establishes the sufficiency direction of the joint ergodicity classification problem for Hardy sequences of polynomial growth: whenever such sequences satisfy the difference and product ergodicity conditions on a system, they are jointly ergodic for that system. This resolves the hard part of the classification for the entire class, without assuming the sequences are pairwise independent. The paper also proves a partial converse and shows by counterexample that the naive equivalence fails in general, since (T^n, $T^{{n+⌊log2 n⌋}}$) is jointly ergodic if and only if T is mixing, while ($T^{{⌊log2 n⌋}}$) is ergodic only for a one-point system.

What carries the argument

The engine of the proof is the family of generalized box seminorms |||·|||^+_{G_1,...,G_s} attached to finitely generated subgroups G_i ⊆ R^ℓ, together with the associated factors Z^+_{G_1,...,G_s} whose defining property is that |||f|||^+_{G_1,...,G_s}=0 exactly when the conditional expectation E(f|Z^+_{G_1,...,G_s}) is zero. The new relative concatenation theorem shows that the intersection of two such factors along a common block of subgroups is contained in the factor along the pairwise sums of the remaining subgroups, a step that lets the smoothing argument merge a preliminary seminorm estimate with an auxiliary estimate. The seminorm smoothing itself is a three-step induction — pinging, ponging, and then concatenating the two seminorms relatively — that upgrades control by box seminorms to control by Host–Kra seminorms of a single transformation once the difference ergodicity condition is in force.

What would settle it

Exhibit a Hardy family a_1,...,a_ℓ ∈ H and a system satisfying the difference and product ergodicity conditions for which the multiple ergodic average E_{n∈[N]} ∏ $T_j^{{⌊a_j(n)⌋}}$ f_j fails to converge in $L^{2}$ to ∏ ∫ f_j dμ; the theorem asserts no such example exists.

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Extended reading notes

Core claim

The paper's main theorem is that the difference and product ergodicity conditions are sufficient for joint ergodicity of Hardy sequences of polynomial growth (Theorem 1.5). More precisely, for a_1,...,a_ℓ ∈ H and a system (X,X,μ,T_1,...,T_ℓ), if each two-step action $T_i^{{⌊a_i(n)⌋}}$ $T_j^{{-⌊a_j(n)⌋}}$ is ergodic and the ℓ-fold product action is ergodic on X^ℓ, then E_{n∈[N]} ∏_{j=1}^ℓ $T_j^{{⌊a_j(n)⌋}}$ f_j converges in $L^{2}$ to ∏ ∫ f_j dμ for all bounded measurable f_j. The machinery used to get there consists of generalized box seminorms and factors, a relative concatenation theorem, and an upgraded three-step seminorm smoothing argument. The converse is not automatic: for (n, n+⌊log2 n⌋) the joint ergodicity of the pair is equivalent to mixing of T, while the single sequence ⌊log2 n⌋ is ergodic only on a one-point system, so joint ergodicity does not imply the difference condition in general.

Load-bearing premise

The proof depends on the existence of generalized box factors Z^+_{G_1,...,G_s} with the property that the generalized box seminorm vanishes exactly when the conditional expectation onto the factor vanishes, a property that in turn rests on a decomposition of multicorrelation sequences whose proof is delegated to a 'straightforward but tedious adaptation' of prior methods.

Editorial extensions

If this is right

  • The difficult direction of the joint ergodicity classification problem is settled for the whole class of Hardy sequences of polynomial growth, with no independence assumption among the sequences.
  • For reasonable Hardy families — those for which every real-linear combination c_i a_i − c_j a_j is either almost rational or logarithmically far from rational polynomials — the difference and product ergodicity conditions become necessary and sufficient for joint ergodicity, resolving the classification for real and fractional polynomials.
  • The construction (T^n, T^{n+⌊log2 n⌋}) shows the unmodified classification problem is false for pathological families, so any valid classification must add a non-pathology hypothesis or switch to weak ergodicity or W-averaging schemes.
  • The new relative concatenation theorem and the general smoothing formalism apply to any ordered family that is good for smoothing, so the proof scheme transfers wherever such initial seminorm estimates are available.

Reading between the lines

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

  • The mixing-versus-ergodic dichotomy for (T^n, T^{n+⌊log2 n⌋}) suggests that the classification breaks precisely when a slowly growing function (like ⌊log n⌋) operates on a timescale comparable to the floor-function errors; a testable conjecture is that replacing log n by any Hardy a with 1 ≺ a ≪ log yields the same dichotomy.
  • The relative concatenation theorem is a structural statement about factors that may have independent uses in additive combinatorics beyond ergodic averages, wherever uniformity-seminorm concatenation is imported.
  • The smoothing argument is phrased abstractly enough that it should extend to tempered functions or other sequence classes that admit initial box-seminorm estimates, as the paper's final conjecture anticipates.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies the joint ergodicity classification problem for Hardy sequences of polynomial growth. Its main theorem (Theorem 1.5) states that if sequences a_1,...,a_ℓ ∈ H satisfy the difference and product ergodicity conditions on a commuting system, then they are jointly ergodic. The proof proceeds through (i) generalized box seminorms and associated factors Z^+_{G_1,...,G_s}, (ii) a relative concatenation theorem for these factors, (iii) a seminorm smoothing upgrade, and (iv) a detailed exponential-sum analysis that converts the difference ergodicity condition into Host-Kra seminorm control. A partial converse is given, together with the counterexample (T^n, T^{n+⌊log_2 n⌋}) being jointly ergodic iff T is mixing, while (T^{⌊log_2 n⌋}) is ergodic only for a one-point system. The paper also proposes weakened classification problems for pathological Hardy sequences.

Significance. If the proof is completed, this would be a substantial advance: it removes the pairwise independence assumptions that were present in the authors' earlier work [16] and resolves the main classification problem for all Hardy sequences of polynomial growth in the difficult direction. The paper also introduces genuinely new tools of independent interest: generalized box factors for finitely generated subgroups of R^ℓ, a relative version of Tao–Ziegler concatenation, and a three-step smoothing argument with a new relative-concatenation step. The counterexample in Theorem 1.10 is concrete and gives an unexpected negative answer to the converse of Problem 1.4. The exponential-sum computations in Sections 5–6 are detailed and appear to be the right technical core for the passage from difference ergodicity to seminorm comparison. The central claims are therefore significant and plausible, but two load-bearing technical steps are not proved in the manuscript.

major comments (3)
  1. [Section 3.1, proof of Theorem 3.3] The proof asserts that a degree-s nilsequence ψ on G = G1×G1×...×Gs×Gs can be approximated in the Følner ℓ²-norm by finite linear combinations of products ψ_s(m_s,m'_s)ψ_{s-1}(m_1,...,m'_{s-1}), and this separation is then used to recast f as an average along G_s+G'_s and to close the argument via Lemma 3.6. No proof or reference is given for this approximation, and it is not a routine density fact: for s=2, G1=G2=Z, the mixed quadratic phase e(α m1 m2) is a degree-2 nilsequence that is orthogonal in the Følner mean to every character, so the claimed separation must be proved by a genuinely nilpotent argument rather than by a standard tensor-product density statement. Since Theorem 3.3 is the base case of the CMS induction behind Corollary 3.2, and since Corollary 3.2 is used in the relative-concatenation step of Proposition 4.11, this gap propagates to Theorems 4.10, 4.5, 1.12, and 1.5. The authors should either prove the separation as a lemma, or replace it by a weaker and explicitly justified approximation statement that still supports the concatenation argument.
  2. [Section 2.2, Proposition 2.15] Proposition 2.15 is the sole input to the converse direction of Proposition 2.13, which establishes property (6) for the generalized box factors Z^+_{G_1,...,G_s}. Its proof is delegated to a 'straightforward but tedious adaptation' of [23,26,38], and the two required properties, (s+1)-weak-anti-uniformity and (s+1)-regularity, are only asserted. The multiparameter extensions of the cited theorems are not stated, and the first paragraph of Section 2.2 itself emphasizes that constructing such factors for arbitrary finitely generated subgroups of R^ℓ is subtle. Because all later results depend on these factors, the manuscript should include either a complete proof of Proposition 2.15 or a precise statement of the needed multiparameter analogues with all hypotheses verified.
  3. [Section 4.3, proof of Proposition 4.11, Step 3] The Step-3 reduction from the bound in (29) to control by | ||f1,1|| |+_{v1,...,vt,u1,...,uk-1,{<uk,u>: u∈R}} uses Corollary 3.2 and Lemma 2.1. This is valid only if the generalized box factors satisfy the weak structure theorem and the relative concatenation theorem. In view of the two previous comments, neither of those ingredients is currently self-contained: Theorem 3.3 depends on the unproved nilsequence separation, and Proposition 2.13 depends on the unproved decomposition Proposition 2.15. The smoothing proof should be rewritten so that every application of Corollary 3.2 is backed by a stated and proved concatenation result, or the paper should explicitly mark these arguments as conditional on the deferred technical lemmas.
minor comments (4)
  1. [Section 2.2, Definition 2.10] The same notation Z^+_{G_1,...,G_s} is used for a subspace of L^∞(µ) and for the corresponding factor of the system; this overloaded notation is occasionally confusing, e.g. in Lemma 3.4 and Corollary 2.12. A separate symbol for the factor, or an explicit convention, would improve readability.
  2. [Section 3.2, Definition 3.7] In part (i) of Definition 3.7, the second tuple uses H'_d while the text elsewhere writes H'_{d'}; the notation should be aligned.
  3. [Section 6.1, Lemma 6.8] The displayed formulas for w and for the exponential sums contain several pairs of unmatched parentheses; a careful rewriting of these expressions would make the computation substantially easier to verify.
  4. [Section 8.1, Proposition 8.6] The proof says that SP implies that the tuple is good for equidistribution. Since this is not immediate and relies on results from [16], a citation or a one-sentence explanation should be added.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity found: the paper's central derivation is built from prior parameter-free results and does not reduce to its own hypotheses by construction.

full rationale

The main derivation chain is: product and difference ergodicity conditions imply good equidistribution via [16, Corollary 12.5] and good seminorm control via Theorem 1.12, after which Theorem 1.11 gives joint ergodicity. The difference ergodicity condition is used as an actual ergodicity input in Proposition 5.1 and Theorem 1.12, not as a relabeled version of the conclusion. The product ergodicity condition is an ergodicity assertion on the product system, which is stronger than the diagonal joint ergodicity conclusion and not identical to it by definition. The paper relies heavily on the authors' prior work [16] for the initial generalized box seminorm estimates, but those estimates are parameter-free and their stated assumptions do not include Theorem 1.5; under the stated review rules this counts as independent support rather than circularity. The proof does contain deferred technical steps, notably the 'straightforward but tedious adaptation' for Proposition 2.15 and the unproved assertion in Theorem 3.3 that a degree-s nilsequence on a product can be approximated by products of lower-degree nilsequences. These are potential proof gaps, and the latter is a genuine load-bearing technical concern, but a missing or false approximation argument is not a circular reduction of the theorem to its inputs. No fitted parameter is relabeled as a prediction, no uniqueness theorem from the same authors is used to force the choice of construction, and no known result is merely renamed in a way that equates input and output.

Assumptions & free parameters 0 free parameters · 6 assumptions · 1 invented entities

The paper is a pure mathematics contribution with no fitted constants or empirical parameters. The main external inputs are standard results in ergodic theory, Hardy field equidistribution, and nilsequence theory, together with the authors' own previous work [16] on generalized box seminorm estimates. The only new construct used as a black box is the family of generalized box factors and its supporting decomposition lemma, whose proofs are only partially written out.

assumptions (6)
  • domain assumption Hardy field functions in H are closed under differentiation, composition, and compositional inversion when defined.
    Defines the class H in Section 1.3; needed for structural results about Hardy sequences used throughout.
  • domain assumption Every finite family in H can be embedded in an ordered Hardy family Q with given sequences in ExtSpanR Q.
    Invoked in the proof of Theorem 1.12 via [43, Lemma A.3] and used to apply the smoothing machinery of Section 4.
  • standard math Boshernitzan's uniform distribution criterion for Hardy sequences (Theorem 1.13).
    Used throughout Sections 5 and 6 for exponential sum limits and equidistribution facts.
  • standard math Herglotz-Bochner spectral theorem for commuting unitary actions.
    Used in Sections 5 and 6 to connect ergodicity of difference sequences to point masses of spectral measures via exponential sums like (4).
  • standard math Tao-Ziegler concatenation theorem for subgroups of Z^ℓ [45].
    The base case of the relative concatenation theorem (Theorem 3.1) and the cited source for the concatenation formalism.
  • ad hoc to paper Proposition 2.15: decomposition of multicorrelation sequences into a nilsequence and a nullsequence.
    Proved only by a sketch referencing [23,26,38]; underpins Proposition 2.13 and the factor theory that all later sections rely on.
invented entities (1)
  • Generalized box factors Z+_{G1,...,Gs}
    purpose: Factors associated with generalized box seminorms for arbitrary finitely generated subgroups of R^ℓ; used to convert seminorm vanishing into conditional expectation vanishing and to run relative concatenation.
    Introduced in Section 2.2 (Definitions 2.8-2.10, Corollary 2.12). They are internal mathematical tools; no external falsifiable prediction is provided. The paper itself notes the subtlety of their construction below Definition 2.10, and their existence theorem (Prop 2.13) rests on a sketched decomposition result.

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Pith. "Pith review of Resolving the joint ergodicity problem for Hardy sequences." pith.science (2026). https://pith.science/paper/RPP4FMYK

@misc{pith2026250620459,
  author       = {Pith},
  title        = {Pith review of: Resolving the joint ergodicity problem for Hardy sequences},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RPP4FMYK}},
  note         = {Machine review of arXiv:2506.20459}
}
read the original abstract

The joint ergodicity classification problem aims to characterize those sequences which are jointly ergodic along an arbitrary dynamical system if and only if they satisfy two natural, simpler-to-verify conditions on this system. These two conditions, dubbed the difference and product ergodicity conditions, naturally arise from Berend and Bergelson's pioneering work on joint ergodicity. Elaborating on our earlier work, we investigate this problem for Hardy sequences of polynomial growth, this time without making any independence assumptions on the sequences. Our main result establishes the "difficult" direction of the problem: if a Hardy family satisfies the difference and product ergodicity conditions on a given system, then it is jointly ergodic for this system. We also find that, surprisingly, the converse fails for certain pathological families of Hardy sequences, even though it holds for all "reasonable" Hardy families. We conclude by suggesting potential fixes to the statement of this problem. New ideas of independent interest developed in this paper include the structure theory of a family of factors generalizing Host-Kra and box factors; a strengthening of Tao-Ziegler's concatenation results; and the most robust extension of a seminorm smoothing argument.

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Forward citations

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