{"id":"e80b7fb0-ce8c-4e1e-b959-20dd88a85c6d","arxiv_id":"2506.20459","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For Hardy sequences of polynomial growth, the paper proves the 'difficult' direction of the joint ergodicity classification conjecture and gives a counterexample showing the converse fails in general.","lead":"This paper resolves a long-standing classification question in ergodic theory for a large family of sparse iterates: when a family of Hardy sequences satisfies the two natural 'difference' and 'product' ergodicity conditions on a dynamical system, it is automatically jointly ergodic. The authors also construct pathological Hardy sequences where the converse fails, showing the original classification statement needs a repair.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Relative concatenation (Thm 3.3) assumes an unproved tensor-product approximation for degree-s nilsequences on product groups; if it fails, the smoothing upgrade behind Thm 1.12 loses its foundation.","rationale":"The reader's weakest assumption concerned Proposition 2.13/2.15, and that concern is real: the existence and structure of the generalized box factors is delegated to prior work and a 'straightforward but tedious adaptation.' I would add a more specific and, in my view, sharper gap: even granting the generalized box factors, the proof of the relative concatenation theorem contains an unproved separation of a degree-s nilsequence into products of nilsequences on coordinate blocks. This is not flagged as an adaptation of an existing theorem, and the minimal mixed-phase example shows why it cannot be treated as a formality. The concern is load-bearing because Theorem 3.3 is not an isolated lemma: it feeds Corollary 3.2, which is used in Step 3 of Proposition 4.11, which produces the upgraded seminorm estimates that Theorem 1.12 needs. If the approximation fails, the smoothing argument has a hole at its base; if it holds, it is exactly the fact the paper should state and prove. I do not see an internal contradiction elsewhere, and much of the surrounding argument is careful; the correct verdict remains conditional pending verification of these structural steps.","tokens_in":68348,"tokens_out":29002,"duration_ms":359332,"concrete_test":"Take s=2, G1=G2=Z, and alpha irrational with bounded partial quotients. Let psi(m1,m'_1,m2,m'_2)=e(alpha m1 m2), viewed as a degree-2 nilsequence. For each fixed K, compute d_K = lim_{N->infty} inf_{sum_{r=1}^K a_r(m1)b_r(m2)} (1/N^2) sum_{m1,m2<=N} |psi - sum_r a_r b_r|^2. If d_K=1 for every K, the tensor-product approximation asserted in Theorem 3.3 is unavailable in the Følner-ell2 norm; alternatively, exhibit an explicit sequence of finite product sums whose average L2 error over boxes tends to 0. This decides whether the reduction in the proof of Theorem 3.3 can be repaired.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing point I find is in the proof of Theorem 3.3: the assertion that a degree-s nilsequence psi on G1 x G1 x ... x Gs x Gs can be approximated in the Følner-ell2 norm by linear combinations of products psi_s(m_s,m'_s) psi_{s-1}(m_1,m'_1,...,m_{s-1},m'_{s-1}). This separation is stated without proof and is not a routine density fact. A degree-s nilsequence on a product group can contain mixed Taylor terms; the minimal example s=2, G1=G2=Z, psi(m1,m2)=e(alpha m1 m2) is a standard degree-2 polynomial nilsequence that is not a product of a function of m1 and a function of m2. Whether it lies in the Følner-ell2 closure of finite sums of such products is exactly what must be checked; for irrational alpha it is actually orthogonal to all characters, so there is no evident approximation by one-variable nilsequences. The proof of Theorem 3.3 uses this separation to recast a dual function with full nilsequence weight as an average along Gs+G'_s, and then to bound the error by the seminorm | ||g|| |+_{...,Gs+G'_s}. Without the separation, the fundamental concatenation argument does not close. Since Theorem 3.3 is the base case of the CMS induction behind Corollary 3.2 and the relative-concatenation step in Proposition 4.11, the smoothing upgrade in Theorem 4.10, hence Theorems 4.5, 1.12, and 1.5, rests on this gap. This is a proof gap rather than a refutation of the theorem, but it is exactly the kind of deferred technical step on which the central claim depends.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":68719,"tokens_out":9131,"duration_ms":113612,"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":[{"comment":"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.","section":"Section 3.1, proof of Theorem 3.3"},{"comment":"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.","section":"Section 2.2, Proposition 2.15"},{"comment":"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.","section":"Section 4.3, proof of Proposition 4.11, Step 3"}],"minor_comments":[{"comment":"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.","section":"Section 2.2, Definition 2.10"},{"comment":"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.","section":"Section 3.2, Definition 3.7"},{"comment":"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.","section":"Section 6.1, Lemma 6.8"},{"comment":"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.","section":"Section 8.1, Proposition 8.6"}],"recommendation":"major_revision","confidential_remarks":"This is a potentially important paper, and the counterexample and exponential-sum sections are strong. My main concern is that the two deferred technical results — Proposition 2.15 and the nilsequence separation in Theorem 3.3 — are exactly the steps on which the new machinery rests. If those are supplied, the paper would likely be suitable for publication in a top journal. The current manuscript is not yet self-contained in those parts."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is the real thing: Theorem 1.5 removes the pairwise independence assumption and nails the difficult direction of the joint ergodicity classification for all Hardy sequences of polynomial growth, and Theorem 1.10's counterexample is genuinely surprising. The generalized box factors, the relative concatenation framework, and the three-step smoothing argument are all new and will likely influence the area even if the present proofs need repair.\n\nBut there is a serious soft spot, and it is exactly where the stress-test note lands. In the proof of Theorem 3.3, the authors assert that any degree-s nilsequence on the product group G1×G1×...×Gs×Gs can be approximated in the Følner-ell2 norm by linear combinations of products ψ_s(m_s,m'_s)·ψ_{s-1}(m_1,m'_1,...,m_{s-1},m'_{s-1}). That is not a routine density fact. For s=2, G1=G2=Z, the nilsequence e(α m1 m2) with irrational α is orthogonal in that norm to every finite sum of products of a function of m1,m1' and a function of m2,m2'. More generally, a degree-s nilsequence on a product group can contain mixed Taylor terms that do not lie in the tensor-product closure. The proof gives no argument for this separation, and it is false in general. Since Theorem 3.3 is the base case for Corollary 3.2 and the relative-concatenation step in Proposition 4.11, the smoothing upgrade and hence Theorems 4.5, 1.12, and 1.5 all rest on this gap.\n\nThe reader's earlier concern about Proposition 2.15 is milder: it is a delegated 'tedious adaptation' of known techniques, which is common in this field, and not obviously wrong. The tensor approximation is the actual load-bearing issue.\n\nThe counterexample, the structure theory, and the smoothing formalism are contributions worth keeping even if Theorem 1.5's proof has a hole. I would send this to a serious referee, but with the explicit request to check whether the concatenation argument can be repaired—perhaps the nilsequence approximation can be replaced by a weaker but valid statement, or the construction of the factors can be adjusted to avoid needing it. As written, it is not ready to accept.\n\nFor citation purposes, I would hold off until the gap is resolved; the counterexample and the tools may still be worth citing independently.","headline":"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.","tokens_in":69284,"tokens_out":3921,"would_cite":false,"duration_ms":50598,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37A44","11B30","28D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["joint ergodicity","Hardy sequences","Hardy fields","multiple ergodic averages","Host-Kra seminorms","box seminorms","seminorm smoothing","concatenation theorems"],"falsifier":"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.","tokens_in":68129,"feed_emoji":"🔄","tokens_out":10845,"duration_ms":94121,"temperature":0.7,"pith_summary":"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.","feed_headline":"Two ergodicity conditions guarantee joint ergodicity","feed_subtitle":"Settles the main direction of the classification for Hardy sequences; the converse fails for a pathological pair.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the generalized box seminorm estimates and the pairwise-independent Hardy-sequence framework that Theorem 4.4 uses as its starting point.","marker":"[16]"},{"why":"Provides the joint ergodicity criteria (Theorem 1.11) and the original seminorm smoothing method that this paper generalizes.","marker":"[30]"},{"why":"Gives the concatenation theorems for box factors along subgroups of Z^ℓ that the new relative concatenation theorem extends.","marker":"[45]"},{"why":"Develops seminorm smoothing for pairwise-dependent integer polynomials, the direct predecessor of the robust smoothing argument used here.","marker":"[28]"},{"why":"Establishes the convergence of exponential sums for Hardy sequences, used in Sections 5 and 6 for the spectral analysis.","marker":"[13]"},{"why":"Gives the equidistribution criterion for Hardy sequences modulo 1 (Theorem 1.13), invoked in the equidistribution and exponential-sum computations.","marker":"[12]"},{"why":"Provides the multiparameter nilsequence decomposition technique that Proposition 2.15 adapts for the multicorrelation sequences.","marker":"[38]"}],"fun_headline_variants":["Two conditions prove joint ergodicity for Hardy sequences","Difference and product ergodicity suffice, converse fails","Joint ergodicity resolved: necessary conditions are sufficient","Hardy sequences: the difficult direction is now proven","Ergodicity classification: sufficiency holds, necessity fails"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Two conditions prove joint ergodicity for Hardy sequences","Difference and product ergodicity suffice, converse fails","Joint ergodicity resolved: necessary conditions are sufficient","Hardy sequences: the difficult direction is now proven","Ergodicity classification: sufficiency holds, necessity fails"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0003,"raw_usage":{"total_tokens":1762,"prompt_tokens":1005,"completion_tokens":757,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":621,"completion_tokens_details":{"reasoning_tokens":680}},"tokens_in":621,"tokens_out":757,"duration_ms":7602,"temperature":1.0,"reasoning_tokens":680,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:47:30.083924+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Frantzikinakis and B","cited_arxiv_id":null,"evidence_quote":"Provides the joint ergodicity criteria (Theorem 1.11) and the original seminorm smoothing method that this paper generalizes."},{"cited_title":"Tao and T","cited_arxiv_id":null,"evidence_quote":"Gives the concatenation theorems for box factors along subgroups of Z^ℓ that the new relative concatenation theorem extends."},{"cited_title":"Frantzikinakis and B","cited_arxiv_id":null,"evidence_quote":"Develops seminorm smoothing for pairwise-dependent integer polynomials, the direct predecessor of the robust smoothing argument used here."},{"cited_title":"Boshernitzan, G","cited_arxiv_id":null,"evidence_quote":"Establishes the convergence of exponential sums for Hardy sequences, used in Sections 5 and 6 for the spectral analysis."},{"cited_title":"Boshernitzan","cited_arxiv_id":null,"evidence_quote":"Gives the equidistribution criterion for Hardy sequences modulo 1 (Theorem 1.13), invoked in the equidistribution and exponential-sum computations."},{"cited_title":"Koutsogiannis","cited_arxiv_id":null,"evidence_quote":"Provides the multiparameter nilsequence decomposition technique that Proposition 2.15 adapts for the multicorrelation sequences."}],"review_version":1}