{"id":"94d3a4d4-3674-4aa7-8f7c-77574fb37ea9","arxiv_id":"2507.15528","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Non-commuting, mixing, zero-entropy transformations can fail double recurrence along p1(n), p2(n) whenever the two polynomials are both linear or both have degree at least two.","lead":"This paper constructs mixing, zero-entropy transformations for which simultaneous returns along two polynomial times never happen, for both linear and higher-degree non-commuting iterates. It completes the classification asked by Frantzikinakis and Host, showing recurrence in general holds only when one iterate is linear and the other has degree at least two.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3.7 depends on Proposition 3.6(c) holding with previous levels evaluated on the long window 0≤j≤2d_k+p_k; if [KS24] only proves a shorter-range independence, Lemma 3.10 and Theorem 1.1 collapse.","rationale":"The reader's weakest assumption correctly identifies the block-independence statement in Proposition 3.6(c) as the linchpin of the new positivity result. My reading confirms that Proposition 3.7's separation of Z(n) and Y(n), and Lemma 3.10's product argument for m(D)>0, need the stronger form of (c) in which previous levels are evaluated on the long window 0≤j≤2d_k+p_k. If [KS24] only establishes the shorter-range version, the proof has a genuine gap and Theorem 1.1 loses its main support. The missing null-set trimming steps in Sections 2.1 and 4 are real but trivial to repair by deleting finitely many null sets; they do not affect the argument's validity. The Gaussian construction for Theorem 1.2 is independent and appears sound, so the paper's conclusions for linear iterates are likely safe. Since the crucial independence statement is explicitly cited from [KS24] and may well be correct, the appropriate verdict remains conditional: accept pending verification of Proposition 3.6(c) as quoted. I therefore recommend no change to the reader's verdict.","tokens_in":18387,"tokens_out":29142,"duration_ms":298113,"concrete_test":"Re-derive Proposition 3.6(c) directly from [KS24, Proposition 2.1] and the construction in [KS24, §2]. Check whether the independence of level k from A_k holds with the previous levels evaluated on 0≤j≤2d_k+p_k, or only on the shorter ranges 0≤j≤2d_l+p_l for l<k. If only the shorter form holds, test whether Lemma 3.10 can still be proved (e.g., by redefining Z(n),Y(n) or using a different independence argument). If m(D)>0 cannot be shown, the concern lands and Theorem 1.1 is unproved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of Theorem 1.1 requires the new positivity result Theorem 3.1(b), proved as Proposition 3.7 via Lemma 3.10. The proof that the events D_k (k≥κ(N)) have positive product measure, and the proof that Z(n) and Y(n) are independent, both invoke Proposition 3.6(c) exactly as stated: for every k, {f̄_k^{(i)}∘T^j : 0≤j≤2d_k+p_k, i=1,2} is independent of A_k = {f̄_l^{(i)}∘T^j : 1≤l<k, 0≤j≤2d_k+p_k}. This is stronger than the 'own-window' independence that a block/tower construction would naturally give. In Proposition 3.7, Z(n) = Σ_{k<K} f_k^{(1)}∘T^n depends on levels k<K evaluated on shifts up to n+d_k+p_k, which for k near K can exceed 2d_k+p_k and reach the long range 2d_K+p_K; Y(n) involves all levels k≥K. The independence of Z(n) and Y(n) therefore requires that the past levels be independent of the tail even when the past levels are evaluated on this longer range. If Proposition 3.6(c) only holds with A_k = {f̄_l∘T^j : 1≤l<k, 0≤j≤2d_l+p_l}, then the induction proving tail-past independence breaks for the long window, and Lemma 3.10's conclusion m(D)>0 no longer follows. That would destroy the positive-measure set C of Proposition 3.5, so Theorem 1.1 has no support. The linear Theorem 1.2 (Section 4) is independent of this block-independence and appears self-contained, but it does not rescue Theorem 1.1. The null-set trimming issue in Section 2.1 is real but easily repaired; the independence statement is the load-bearing point.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs counterexamples to double recurrence for non-commuting, mixing, zero-entropy measure-preserving transformations. Theorem 1.1 handles the case where both iterates are injective integer polynomials of degree at least 2 vanishing at 0, and Theorem 1.2 handles the case where both iterates are linear (c n and d n). Together with an earlier positive result of Frantzikinakis and Host for one linear and one higher-degree iterate, the paper claims a complete answer to the double-recurrence question in the zero-entropy non-commuting setting. The higher-degree construction uses a 2-dimensional local central limit theorem cocycle from [KS24] and a new positivity result for the range of the cocycle; the linear construction uses Gaussian automorphisms built from singular measures with polynomial Fourier decay.","tokens_in":18839,"tokens_out":23451,"duration_ms":207387,"significance":"If the proofs are correct, the results settle the remaining cases of a natural question of Frantzikinakis and Host and are therefore significant for the field of non-commuting multiple recurrence. The Gaussian construction in Section 4 is elegant and largely self-contained, relying only on standard facts about Gaussian automorphisms and the known existence of singular measures with polynomial Fourier decay. The new positivity result Theorem 3.1(b) is of independent interest. The paper is clearly written and the main ideas are well motivated, with the simple n,n case presented first as a baby model.","major_comments":[{"comment":"The displayed inequality (1 - α_k^2)^{p_k + 2N} ≥ exp(-α_k^2(p_k + 2N)) is false for 0 < α_k < 1, since log(1-x) < -x for x > 0. The subsequent estimate exp(-α_k^2(p_k+2N)) ≥ exp(-2/p_k) therefore does not follow from the previous line. The lemma is repairable: using the standard bound (1-x)^m ≥ exp(-2mx) for x ≤ 1/2 yields exp(-4/p_k) instead, and the product over k of exp(-4/p_k) still converges because ∑ 1/p_k < ∞. Nevertheless, the proof as written contains a false inequality in a key estimate and must be corrected.","section":"§3.2, Lemma 3.10"},{"comment":"The proof that the set A satisfies A ∩ T^{-n}A ∩ S^{-n}A = ∅ for all n ∈ N is incomplete. From the maximality of M, one obtains that for m > M with m ≤ N, the set D ∩ T^{-m}D ∩ S^{-m}D has measure zero, not that it is empty. Therefore the containment A ∩ T^{-n}A ∩ S^{-n}A ⊂ D ∩ T^{-(n+M)}D ∩ S^{-(n+M)}D only shows that the intersection is contained in a null set for n+M ≤ N, not that it is empty. The argument can be fixed by removing the null set ⋃_{m=M+1}^{N} (D ∩ T^{-m}D ∩ S^{-m}D) from D before defining A, but this step is missing. The same gap appears in the Gaussian proof in Section 4 and affects the proof of Theorem 1.2 in both cases.","section":"§2.1 and §4, trimming argument"},{"comment":"The proof of the new positivity result Proposition 3.7 and of Lemma 3.10 relies critically on Proposition 3.6(c) as quoted from [KS24], where the previous levels {f̄_l, l<k} are evaluated on the long window 0 ≤ j ≤ 2d_k + p_k. This is a stronger independence statement than the 'own-window' independence that a natural block construction would give. If [KS24] only proves the weaker statement, then the independence of the sets {D_k} in Lemma 3.10 and the independence of Z(n) and Y(n) in Proposition 3.7 fail, and the proof of Theorem 1.1 collapses. The authors should either prove Proposition 3.6(c) in this paper or give a precise reference and verify that [KS24] indeed contains exactly this statement. This is the load-bearing point for Theorem 1.1.","section":"§3.2, Proposition 3.6(c) and Proposition 3.7"}],"minor_comments":[{"comment":"The polynomial ring is denoted Z(t) in two places; this should be Z[t], since Z(t) conventionally denotes the field of rational functions.","section":"Introduction"},{"comment":"The symbol B is used both for the product sigma-algebra B = D^{⊗3} and for the set B = (Y × [1]_0)^{⊗3}. This overloading is confusing in the proof of Theorem 1.2 (c=d=1); please use different letters.","section":"§2.1"},{"comment":"The notation K(n) is used in the sentence 'Z(n) is a function of {f̄_k^{(1)}∘T^j : 1 ≤ k < K(n), 0 ≤ j ≤ 2d_K + p_K}' even though K was defined earlier as K(N); also the claim that all shifts appearing in Z(n) lie in the stated range is not justified and should be spelled out.","section":"§3.2, Proposition 3.7"},{"comment":"The symbol K(N) is reused with two different meanings: in Proposition 3.7 it is the smallest integer with 2N < p_K, while in Lemma 3.10 it is the smallest integer larger than κ with 2p_K < d_K. Please use different names to avoid ambiguity.","section":"§3.2, Lemma 3.10"},{"comment":"The function f is defined as f = 2g in Section 3 and then redefined in Section 3.2 as the [KS24] LCLT function. Since scaling by 2 preserves the range-distinctness property needed for Theorem 3.1(b), the two uses are compatible, but this should be stated explicitly to avoid confusion.","section":"§3, Section 3.2"},{"comment":"The sentence 'Note that as p_k + 2N < 2p_k there is no contradiction in the definition of D_k' is terse. What is actually needed is d_k > p_k + 2N, which follows from the defining condition 2p_K < d_K; please clarify.","section":"§3.2, Lemma 3.10"}],"recommendation":"major_revision","confidential_remarks":"The main risk is the heavy reliance on the self-cited preprint [KS24] for Proposition 3.6(c), which is exactly the strong independence required for the new positivity result. Since the authors are also the authors of [KS24], they are in a position to verify the statement and should be asked to do so explicitly. The false inequality in Lemma 3.10 and the trimming gap are local but must be fixed before the paper can be accepted. The Gaussian section is in good shape modulo the same trimming issue."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline is that this is a genuine advance. Theorems 1.1 and 1.2 together with the FH23 positive result give the full classification for injective polynomial iterates vanishing at zero: recurrence holds exactly when one polynomial is linear and the other has degree at least two, and fails in the two-linear and two-high-degree cases. The construction for degree at least two is the substantive new work, and the cocycle-range positivity result in Section 3.2 is a real new tool, not a repackaging of [KS24]. The Gaussian treatment of the linear case in Section 4 is clean, self-contained, and handles arbitrary nonzero c,d via roots of Gaussian automorphisms.\n\nI agree with the reader that the central arguments are coherent. I also checked the stress-test concern about Proposition 3.7 and do not think it lands. The needed ranges work out: Z(n) is a function of the previous levels on [0,2d_K+p_K], and for each tail level k≥K the shifts used by Y(n) lie inside [0,2d_k+p_k]. So Proposition 3.6(c), exactly as stated, gives the required independence by induction over tail levels. The proof is terse and would benefit from spelling that induction out, but the independence claim is supported by the quoted block-independence.\n\nThe real soft spots are two fixable gaps. First, Lemma 3.10 states (1-alpha^2)^L >= exp(-alpha^2 L), which has the wrong direction; the valid bound with an extra factor of 2 still gives a summable product, so nothing breaks. There is also a missing factor of 2 in the exponent for D_k, again harmless. Second, in Section 2.1 and again in Section 4, the passage from positive-measure D to a set A with literally empty intersections is too quick: the maximality of M only gives measure zero, not emptiness, when n+M is still small. The standard fix is to remove the countable union of the measure-zero intersections from A; this is a routine repair.\n\nThe reliance on [KS24] is substantial, especially for the 2D LCLT and the block-independence proposition. That is not circular, because the new positivity result is proved here, and the Gaussian section is independent. But referees should check that [KS24] actually proves Proposition 3.6(c) in the stated long-window form, since the paper explicitly depends on it.\n\nWho is this for? Ergodic theorists and ergodic Ramsey theorists working on non-commuting multiple recurrence. It deserves a serious referee and, with the small repairs above, should be publishable in a good journal. I would send it to peer review rather than desk reject.","headline":"This paper likely closes the Frantzikinakis–Host double-recurrence question for non-commuting zero-entropy systems, and the main construction survives a close check; the two gaps I found are fixable without changing the statements.","tokens_in":19385,"tokens_out":10750,"would_cite":true,"duration_ms":113346,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["28D05","37A05","37A50","37A30","60F05","60G10","60G15"],"pacs":[],"model":"deepseek-v4-flash","headline":"For injective integer polynomials that vanish at the origin and have equal degree (both linear or both degree at least two), there exist mixing, zero-entropy, non-commuting transformations $T$ and $S$ and a positive-measure set $A$ whose…","keywords":["Multiple recurrence","Non-commuting transformations","Zero entropy","Local central limit theorem","Polynomial iterates","Mixing transformations","Gaussian automorphisms","Polynomial Fourier decay"],"falsifier":"Compute the measure of the event $0<S_1(f)<\\cdots<S_{2N}(f)$ in the lexicographic order for the explicit block cocycle from Proposition 3.6 with $N=1$; Proposition 3.7 declares this measure positive, and if it were zero the permutation construction in Lemma 3.4 would collapse.","tokens_in":18187,"feed_emoji":"🔁","tokens_out":15466,"duration_ms":148433,"temperature":0.7,"pith_summary":"The paper proves that double recurrence can fail completely for non-commuting, mixing, zero-entropy transformations whenever the two polynomial iterates have the same degree: both linear, or both of degree two or higher. For any injective integer polynomials $p_1,p_2$ with $p_1(0)=p_2(0)=0$, the authors construct transformations $T,S$ and a set $A$ of positive measure such that $A \\cap T^{-p_1(n)}A \\cap S^{-p_2(n)}A = \\varnothing$ for every $n \\in \\mathbb{N}$. Together with the positive result already known for one linear and one higher-degree iterate, this settles the motivating question completely and marks the mixed-degree pair as the only case where double recurrence is guaranteed. The higher-degree construction uses a skew product over a zero-entropy base with a two-dimensional cocycle satisfying a lattice local central limit theorem, while the linear case uses Gaussian automorphisms with singular spectral measures.","feed_headline":"Same-degree polynomial iterates defeat double recurrence","feed_subtitle":"Their construction makes the triple intersection empty for every n, completing the classification of polynomial iterates.","key_machinery":"For the higher-degree theorem, the load-bearing object is a two-dimensional cocycle $S_n(f)$ built from i.i.d. blocks, satisfying a lattice local central limit theorem and, newly, a distinct-visits property: for every $N$, the increments $\\{S_{j+N}(f)-S_N(f): -N\\le j\\le N\\}$ take $2N+1$ distinct lattice points on a set of positive measure (Proposition 3.7). The proof obtains this by forcing a lexicographically increasing run in the first coordinate, using block independence on the long range $0\\le j\\le 2d_k+p_k$. Around this cocycle the authors form a skew product $\\tilde T$ over a zero-entropy base, then define a permutation $\\pi_y:\\mathbb{Z}^2\\to\\mathbb{Z}^2$ that transports the $p_2$-return values $S_{p_2(n)}(f)(y)$ to the $p_1$-return values for every $n$ in a density-one set $K_y$; conjugating $\\tilde T$ by $\\pi_y$ yields $\\tilde S$, and in a $k$-fold product the two dynamics flip the same coordinate bit in opposite directions. For the linear theorem, the machinery is instead a unitary reflection $W = P - (I-P)$ on the Gaussian space, where $P$ is projection onto the span of the Gaussian variable; the transformation $S$ has Koopman operator $V = WUW$, the intersection measure decays like $|\\hat\\sigma(n)|^{2k}$ with polynomial Fourier decay, and $k$ is chosen so that the probabilities are summable.","core_discovery":"On the paper's own terms, the discovery is that the obstruction to double recurrence in the non-commuting zero-entropy world is degree parity: if both iterates are linear or both have degree at least two, the recurrence conclusion of the commuting polynomial theorem is false in the strongest sense. Theorem 1.1 asserts that for injective $p_1,p_2 \\in \\mathbb{Z}[t]$ with $p_1(0)=p_2(0)=0$ and $\\deg p_i \\ge 2$, there exist mixing zero-entropy transformations $T,S$ and a set $A$ with $\\mu(A)>0$ such that $A \\cap T^{-p_1(n)}A \\cap S^{-p_2(n)}A = \\varnothing$ for all $n \\in \\mathbb{N}$; Theorem 1.2 is the same statement for $p_i(n)=c_i n$ with nonzero integers $c_i$. The constructions are explicit: a skew product driven by a $\\mathbb{Z}^2$-valued cocycle with an i.i.d.-block structure, conjugated by a coordinate permutation that swaps the two polynomial-return sets, and, for linear iterates, $k$-fold products of Gaussian automorphisms whose spectral measures have polynomial Fourier decay and which possess roots of all orders. The conclusion is a complete answer to the question posed in [FH23]: the recurrence theorem for the pair $(n,p(n))$ with $\\deg p \\ge 2$ marks exactly the boundary of the phenomenon.","pith_inferences":["The long-range block-independence property behind Proposition 3.7 should transfer to other cocycle questions: whenever a zero-entropy system carries a cocycle whose blocks are independent over ranges longer than the block length, bounded windows of partial sums visit distinct lattice points with positive probability, which is exactly the input needed for further non-convergence examples.","Because the Gaussian construction gives roots of all orders, the same method likely handles linear iterates with rational slopes, or even pairs of affine sequences, by reparametrizing the embedded flow; the paper only states integer coefficients $c,d$.","A natural stress test is to push the same equal-degree construction to $d$-tuples of polynomials: the $k$-fold product trick suggests multiple recurrence fails for any tuple of injective equal-degree polynomials, and the mixed-degree boundary from [FH23] could be explored in higher-order averages."],"forward_implications":["For every injective integer polynomial pair $p_1,p_2$ with $p_1(0)=p_2(0)=0$ and $\\deg p_1,\\deg p_2\\ge2$, there exist mixing, zero-entropy transformations $T,S$ and a positive-measure set $A$ with $A\\cap T^{-p_1(n)}A\\cap S^{-p_2(n)}A=\\varnothing$ for every $n\\in\\mathbb{N}$.","For every pair of nonzero integers $c,d$, the same total failure holds for the linear iterates $cn$ and $dn$, extending the previously known $n,n$ case.","Together with the positive result in [FH23], the motivating question is now answered completely: the only pair type for which double recurrence is guaranteed in this setting is one linear iterate together with one higher-degree iterate.","The constructed counterexamples are mixing and have zero entropy, so the failure is not an artifact of poor mixing or positive entropy; it is intrinsic to non-commuting deterministic pairs."],"supporting_citations":[{"why":"It supplies the two-dimensional local central limit theorem and the i.i.d.-block cocycle construction whose independence properties are sharpened in Proposition 3.7.","marker":"[KS24]"},{"why":"It proves the one-dimensional lattice local central limit theorem used in the $n,n$ construction and in the mixing proof for the skew product.","marker":"[KV22]"},{"why":"It provides the positive double-recurrence theorem for one linear and one higher-degree iterate, and it poses the question the paper answers.","marker":"[FH23]"},{"why":"It establishes singular measures on the circle with polynomial Fourier decay, the spectral ingredient for the Gaussian linear counterexamples.","marker":"[Kau80]"},{"why":"It contributes the non-convergence construction ideas on which the Gaussian counterexamples in Section 4 are built.","marker":"[Aus25]"},{"why":"It handles the special $c=d=1$ linear case and supplies the Cartesian-power trick used to amplify the Gaussian estimate.","marker":"[Ryz24b]"},{"why":"It records the Gaussian automorphism properties, including roots of all orders, needed to pass from the $n,n$ case to general linear iterates.","marker":"[JRDLR23]"},{"why":"It is cited for the fact that Gaussian automorphisms with singular spectral measure have zero entropy.","marker":"[Pin60, dlR93]"}],"fun_headline_variants":["Degree parity blocks double recurrence in non-commuting maps","Same-degree iterates defeat double recurrence for zero entropy","Double recurrence fails for non-commuting iterates sharing a degree","Matching polynomial degrees block double recurrence in zero entropy","Equal-degree iterates create counterexample to double recurrence"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the block independence of the two-dimensional cocycle holds over the long range $0\\le j\\le 2d_k+p_k$ (and, for the linear theorem, that singular spectral measures with polynomial Fourier decay exist); if either fails, the positive-measure set witnessing empty intersections may not exist.","fun_headline_variants_meta":{"raw":{"variants":["Degree parity blocks double recurrence in non-commuting maps","Same-degree iterates defeat double recurrence for zero entropy","Double recurrence fails for non-commuting iterates sharing a degree","Matching polynomial degrees block double recurrence in zero entropy","Equal-degree iterates create counterexample to double recurrence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.002375,"raw_usage":{"total_tokens":9120,"prompt_tokens":899,"completion_tokens":8221,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":515,"completion_tokens_details":{"reasoning_tokens":8141}},"tokens_in":515,"tokens_out":8221,"duration_ms":60400,"temperature":1.0,"reasoning_tokens":8141,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:31:43.712289+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the measure of the event $0<S_1(f)<\\cdots<S_{2N}(f)$ in the lexicographic order for the explicit block cocycle from Proposition 3.6 with $N=1$; Proposition 3.7 declares this measure positive, and if it were zero the permutation construction in Lemma 3.4 would collapse.","supporting_citations":[],"review_version":1}