{"id":"c44efbca-86bb-4539-91b8-8bd0d1a0e080","arxiv_id":"2607.09358","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For rationally independent nonlinear polynomials, the set of n giving nearly independent intersections of a positive-density set has positive lower IP_rat density.","lead":"The paper proves that large intersections for rationally independent nonlinear polynomials hold with positive lower density inside every IP set generated by a rational-spectrum sequence. This strengthens classical multiple-recurrence theorems by combining the large-intersection property with IP structure restricted to rational spectrum.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the nonlinearity hypothesis as the only substantive limitation and notes that it is openly converted into a conjecture. Because the proved theorems never claim more than the nonlinear setting, that limitation does not undermine the central claim. The supporting lemmas (IP mean ergodic theorem, IP van der Corput, density of vertical characters, equidistribution of polynomial orbits on connected components) are either cited from prior work or proved in full; the PET induction and the reduction to the rational Kronecker factor follow the classical pattern without gaps. Consequently no load-bearing concern arises that would justify altering the ACCEPT verdict. The suggested concrete test simply reconfirms the induction base that the paper already treats as standard.","tokens_in":25205,"tokens_out":521,"duration_ms":5279,"concrete_test":"Verify the base case of Proposition 3.8 for the single nonlinear polynomial p(n)=n^{2} by direct application of the IP van der Corput lemma (Lemma 2.13) and the IP mean-ergodic theorem (Theorem 2.11) on the Host–Kra factor Z_1; confirm that the resulting limit vanishes whenever E(f|Z_1)=0. If this elementary check holds, the induction that underpins Theorem 3.9 is secure for the nonlinear regime.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 1.7 / Corollary 1.8) is restricted to strictly nonlinear rationally independent polynomials, and the paper’s own reduction (Theorem 3.9) and characteristic-factor identification (Theorem 6.2) both use that hypothesis in an essential way (the PET step that produces a standard family after adding a high-degree q, and the subsequent density argument that forces H = (G°)^k). The authors flag the restriction explicitly and convert the linear-mixed case into Conjecture 1.9. Within the stated nonlinear regime the classical Host–Kra / Leibman / Frantzikinakis–Kra toolkit is applied carefully; the IP_rat mean-ergodic theorem, the IP van der Corput lemma, and the pointwise nilmanifold formula are all established before they are invoked. No hidden circularity or unjustified approximation appears. The reader’s weakest-assumption note is therefore an accurate description of the paper’s own boundary rather than a soft spot inside the proved statement.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper proves that for any invertible measure-preserving system, any set A of positive measure, and any family of rationally independent nonlinear integer polynomials p1,...,pk with pi(0)=0, the set of n for which the multiple intersection measure exceeds µ(A)^{k+1}-ε has positive lower IP_rat-density (Theorem 1.7). The combinatorial counterpart (Corollary 1.8) follows by the Furstenberg correspondence principle. The argument proceeds by establishing that Host–Kra factors Z_{k-1}(X) are IP_rat-characteristic for standard nonlinear polynomial systems (Proposition 3.8 via PET induction and an IP van der Corput lemma), reducing general nonlinear systems to the standard case (Theorem 3.9), proving a pointwise limit formula for continuous functions on nilmanifolds along rational IPs (Theorems 4.5 and 5.9, generalizing Leibman), and finally identifying the rational Kronecker factor as the minimal IP_rat-characteristic factor (Theorem 6.2). The large-intersection claim is then obtained by a weighted averaging argument that exploits the positivity of the density of n with all polynomial values congruent to 0 mod r (Lemma 7.2).","tokens_in":25439,"tokens_out":820,"duration_ms":8673,"significance":"The result simultaneously upgrades the Frantzikinakis–Kra large-intersection theorem for polynomials from syndeticity to positive lower IP_rat-density and extends the linear IP_rat results of Kra–Shalom to the nonlinear setting. The pointwise convergence theorem for polynomial sequences along rational IPs (Theorem 5.9) is of independent interest and supplies a clean generalization of Leibman’s classical formula. Within the stated nonlinear regime the proofs are complete and rest on standard, non-circular black boxes (Host–Kra structure, Leibman equidistribution, the authors’ earlier IP mean-ergodic theorem). The restriction to strictly nonlinear polynomials is flagged explicitly and converted into a precise conjecture (Conjecture 1.9), so the paper does not overclaim.","major_comments":[],"minor_comments":[{"comment":"Abstract and title use “IPrat” / “IP_rat” inconsistently with the body; a single notation should be fixed throughout.","section":null},{"comment":"Definition 2.4 of IP as a multiset is non-standard; a short remark clarifying that multiplicities do not affect the averages would help readers.","section":null},{"comment":"In the proof of Lemma 3.5 the appeal to Sperner’s theorem is correct but could be replaced by a simpler double-counting argument; either way a one-line reference or expansion would improve readability.","section":null},{"comment":"Page 17, display after “we have that H=(G°)^k”: the product of integrals is written with connected components X_{x_j,i}; a brief reminder that these are the components of the original system would avoid a momentary ambiguity.","section":null},{"comment":"Several arXiv preprints are cited as 2025/2026; once published versions appear the bibliography should be updated.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is a solid, carefully written contribution that sits comfortably in the Host–Kra / Leibman / Frantzikinakis–Kra tradition. The nonlinear restriction is genuine but honestly acknowledged; I see no reason to delay acceptance for the mixed-linear conjecture. Fit for a strong dynamics journal is clear."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The paper delivers exactly what the abstract promises: for rationally independent nonlinear integer polynomials vanishing at 0, the large-intersection set has positive lower density inside every rational-spectrum IP (Theorem 1.7 / Corollary 1.8). That simultaneously strengthens Frantzikinakis–Kra (syndeticity) and Bergelson–McCutcheon (IP* positivity) under the spectral restriction already known to be necessary in the linear case.\n\nWhat is genuinely new is the IP_rat-characteristic-factor machinery and the pointwise convergence theorem for polynomial sequences along rational IPs on nilmanifolds (Theorem 5.9), which generalizes Leibman. The argument is classical PET + IP van der Corput + Host–Kra + Leibman equidistribution, carefully rewritten for IP_rat averages; every step is spelled out and the earlier mean-ergodic theorem is used as a black box whose statement does not contain the target claim. The reduction to the rational Kronecker factor (Theorem 6.2) is clean once one is inside the nonlinear regime.\n\nThe only real boundary is the strict nonlinearity hypothesis. Both the PET reduction that produces a standard family and the density argument that forces H = (G°)^k use it essentially; the authors flag this and convert the mixed-linear case into Conjecture 1.9. That is honest bookkeeping, not a hidden gap inside the proved statement. No circularity, no free parameters, citations look standard.\n\nThis is for people who already work in ergodic Ramsey theory or additive combinatorics via characteristic factors. It is not a breakthrough that rewrites the field, but it is a clean, usable advance that a serious referee should see. I would accept it for peer review and expect it to appear after ordinary polishing.","headline":"Solid upgrade of Frantzikinakis–Kra large intersections to positive lower IP_rat density for nonlinear independent polynomials, with a clean new pointwise IP_rat formula on nilmanifolds.","tokens_in":26049,"tokens_out":468,"would_cite":true,"duration_ms":5250,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37A30","37A05","11B30","28D05"],"pacs":[],"model":"grok-4.5","headline":"Rationally independent nonlinear polynomials produce large intersections along every rational-spectrum IP set.","keywords":["IP sets","rational spectrum","polynomial recurrence","large intersections","characteristic factors","Host–Kra factors","rational Kronecker factor","nilmanifolds"],"falsifier":"Exhibit a single linear polynomial together with one or more nonlinear polynomials that remain rationally independent, and a system in which the corresponding multiple averages along some rational-spectrum IP do not vanish when a function orthogonal to every Host–Kra factor is inserted; this would refute the natural extension of the characteristic-factor theorem.","tokens_in":26100,"feed_emoji":"∞","tokens_out":798,"duration_ms":8225,"temperature":0.7,"pith_summary":"This paper proves that if you take a set of natural numbers that is dense in the usual sense, and you look for simultaneous returns along several rationally independent nonlinear polynomials, those returns are not only positive but nearly as large as independent chance would predict, and they occur with positive relative density inside every IP set generated by a sequence whose spectrum is rational. The same statement holds in the dynamical setting: for any measure-preserving system and any positive-measure set A, the multiple intersections A ∩ T^{p1(n)}A ∩ … ∩ T^{pk(n)}A stay larger than µ(A)^{k+1}-ε on a set of n that has positive lower IP_rat-density. The authors reach this by first showing that Host–Kra factors control the polynomial averages taken along rational IPs, then refining the characteristic factor all the way down to the rational Kronecker factor, and finally obtaining a pointwise limit formula for continuous functions on nilmanifolds that generalizes Leibman’s classical result.","feed_headline":"Polynomial large intersections fill every rational IP set","feed_subtitle":"Near-optimal multiple returns along nonlinear polynomials occur densely inside every rational-spectrum IP","key_machinery":"IP_rat-characteristic factors: a factor Y of a system X is IP_rat-characteristic for a polynomial system P if the multiple ergodic averages along any rational-spectrum IP vanish whenever one of the functions is orthogonal to Y. The paper proves that Host–Kra factors of finite order, and ultimately the rational Kronecker factor itself, are IP_rat-characteristic for nonlinear rationally independent polynomials.","core_discovery":"For every invertible measure-preserving system, every positive-measure set A, every family of rationally independent nonlinear integer polynomials p1,…,pk vanishing at zero, and every ε>0, the set of n for which µ(A ∩ T^{p1(n)}A ∩ … ∩ T^{pk(n)}A) exceeds µ(A)^{k+1}-ε has positive lower IP_rat-density. The combinatorial translation is that the same large-intersection set intersects every IP generated by a rational-spectrum sequence with positive relative density along every Følner sequence.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Rational IPs always meet near-optimal polynomial intersections","Large multiple returns along independent polynomials dense in every rational IP","Nonlinear polynomial intersections fill all rational-spectrum IPs","Every rational IP hits large intersections of polynomial shifts","IP_rat sets densely capture near-optimal polynomial multiple returns"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The reduction that produces a standard polynomial family and identifies the rational Kronecker factor as characteristic works only when every polynomial is strictly nonlinear; the presence of even one linear polynomial breaks the argument.","fun_headline_variants_meta":{"raw":{"variants":["Rational IPs always meet near-optimal polynomial intersections","Large multiple returns along independent polynomials dense in every rational IP","Nonlinear polynomial intersections fill all rational-spectrum IPs","Every rational IP hits large intersections of polynomial shifts","IP_rat sets densely capture near-optimal polynomial multiple returns"]},"model":"grok-4.5","effort":"low","cost_usd":0.005028,"raw_usage":{"total_tokens":1326,"prompt_tokens":732,"num_sources_used":0,"completion_tokens":80,"cost_in_usd_ticks":50280000,"prompt_tokens_details":{"text_tokens":732,"audio_tokens":0,"image_tokens":0,"cached_tokens":0},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":514,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":732,"tokens_out":80,"duration_ms":5261,"temperature":1.0,"reasoning_tokens":514,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-13T03:36:37.354017+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a single linear polynomial together with one or more nonlinear polynomials that remain rationally independent, and a system in which the corresponding multiple averages along some rational-spectrum IP do not vanish when a function orthogonal to every Host–Kra factor is inserted; this would refute the natural extension of the characteristic-factor theorem.","supporting_citations":[],"review_version":1}