{"id":"c616cc44-28e9-4880-9186-21f3f954a86f","arxiv_id":"2607.10051","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.5,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Simultaneous popular polynomial differences hold over large F_p for linearly independent zero-constant polynomials, but fail for paired 3-APs over F_p^n as n grows.","lead":"The paper proves that over large finite fields, a single nonzero difference d can make many polynomial configurations simultaneously popular in any dense set. It also shows this simultaneous popularity fails for certain related arithmetic progressions in high-dimensional spaces over fixed fields.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"The simultaneous lower bound for all 2^k subconfigurations rests on linear independence and vanishing constant terms of the P_i; these hypotheses are load-bearing for the existence of a common popular d over large F_p.","rationale":"The reader correctly isolates the structural hypotheses (linear independence + zero constants) as the weakest assumption supporting the simultaneous claim. Because only the abstract is available, no further internal inconsistency or quantitative gap can be audited; the same hypotheses remain the single most load-bearing point. The proposed concrete test directly probes necessity of independence over the same field F_p that appears in the positive statement, and would either confirm or remove the concern. Consequently the UNVERDICTED status and low confidence are left unchanged.","tokens_in":2365,"tokens_out":615,"duration_ms":18416,"concrete_test":"Fix the dependent pair P={t,2t} and, for a sequence of large primes p, search (by exhaustive or random sampling of dense sets, or by adapting the paper’s own F_p^n construction) for A⊂F_p of density α≈1/2 such that max_{d≠0} min{ E 1_A(x)1_A(x+d), E 1_A(x)1_A(x+2d) } falls below α^2-ε for some fixed ε>0. If such A exist for infinitely many p, linear independence is necessary and the concern lands; if every dense A still admits a common popular d, the hypothesis can be relaxed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central positive claim asserts a single nonzero d that simultaneously realises the density lower bound α^{1+∑ω_i}-ε for every subconfiguration indexed by ω∈{0,1}^k. The abstract explicitly requires the fixed collection P to be linearly independent over Z and to satisfy P_i(0)=0. These conditions are the least secure point: if independence fails (e.g., one polynomial is an integer multiple of another), the associated difference sets become linearly related, so the popular d for one configuration can systematically avoid popularity for another. The paper’s own negative result (failure of simultaneous popularity for d and 2d over F_p^n) already demonstrates that such conflicts occur once freeness is lost. Without the full argument it is impossible to see precisely where independence is invoked—most likely in a counting lemma, Gowers-norm inverse theorem, or Fourier-analytic equidistribution step that treats the map d↦(P_1(d),…,P_k(d)) as free—but the claim as stated collapses if that step requires freeness. Zero constant terms are likewise essential to keep the configurations through the origin and avoid trivial affine shifts that would destroy the density-increment or energy-increment arguments for large p.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript claims a simultaneous popular-difference theorem for polynomial configurations over prime fields: for any fixed collection of linearly independent polynomials P_1,...,P_k in Z[t] with zero constant terms, every ε>0, all large primes p, and every A⊆F_p of density α, there exists nonzero d such that the configuration density E_x 1_A(x)∏ 1_A(x+P_i(d))^{ω_i} is at least α^{1+∑ω_i}-ε simultaneously for every ω∈{0,1}^k. A complementary negative result asserts that over F_p^n (p fixed, n→∞) simultaneous popularity of both d and 2d for three-term APs fails: there exist sets of density 1/2+o(1) for which the min of the two configuration densities is at most 1/8-c for some c>0 independent of n.","tokens_in":2632,"tokens_out":909,"duration_ms":13048,"significance":"If the positive theorem holds as stated, it is a clean and natural strengthening of Green's popular-difference theorem to a simultaneous multi-configuration setting for polynomial progressions, of clear interest in additive combinatorics over finite fields. The negative result supplies a sharp limitation by exhibiting an explicit conflict between two linearly related configurations, showing that freeness hypotheses cannot be dropped casually. The claims are stated with external density benchmarks (α^3, α^{1+∑ω_i}, 1/8) and are in principle falsifiable; the simultaneous formulation and the density-1/2 counterexample construction are the main contributions.","major_comments":[{"comment":"Only the abstract is available for review, so the derivations, error estimates, counting lemmas, and the explicit construction of the density-1/2 counterexample over F_p^n cannot be checked. The central positive claim and the sharpness statement therefore remain unverified; a full assessment of soundness is impossible on the present material.","section":null},{"comment":"The positive theorem (as stated in the abstract) takes linear independence of the P_i over Z and vanishing constant terms as hypotheses. These are load-bearing: the paper's own negative result already shows that simultaneous popularity fails once freeness is lost (d versus 2d). The manuscript must make explicit where independence and P_i(0)=0 enter the argument (e.g., equidistribution of d↦(P_1(d),...,P_k(d)), Gowers-norm control, or Fourier analysis) and should indicate whether either hypothesis can be relaxed.","section":null},{"comment":"The negative result asserts a uniform gap c>0 below 1/8 for the min of the two 3-AP densities, for sets of density 1/2+o_n(1). Without the construction or the quantitative estimates, it is unclear whether the o_n(1) and the constant c are robust, or whether the same obstruction appears already in F_p (rather than only in high-dimensional F_p^n). This gap is essential to the claim that the simultaneous strengthening of Green is false in that regime.","section":null}],"minor_comments":[{"comment":"The abstract is clearly written and the statements are easy to parse; once the full text is available, ensure that the main theorems are numbered and that the dependence of the 'sufficiently large p' threshold on ε, k and the degrees of the P_i is recorded explicitly.","section":null},{"comment":"A brief comparison with existing popular-difference or popular-polynomial results (beyond Green) would help place the simultaneous bound and the 1/8-c obstruction in context.","section":null}],"recommendation":"uncertain","confidential_remarks":"Full text was not supplied (abstract-only review). I cannot responsibly recommend accept/minor/major/reject until the proofs and the counterexample construction are available. The abstract itself is coherent and the claims sit comfortably inside the expected shape of popular-difference theorems; the linear-independence hypothesis is the natural stress point and is already flagged by the authors' own negative result. Please supply the manuscript body for a proper report."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing to know is that this paper upgrades Green's popular-difference theorem to a simultaneous polynomial version over large prime fields and then shows the phenomenon has sharp limits. For any fixed linearly independent family of zero-constant polynomials P1,...,Pk, one nonzero d works at once for every subconfiguration: the density lower bound α^{1+∑ω_i}-ε holds simultaneously for all 2^k choices of which factors to include. They also give a density-1/2 set in F_p^n (n large) where no d makes both d and 2d popular for ordinary 3-APs, so the simultaneous strengthening fails once freeness is lost.\n\nThat combination is new and useful. Green's original result is classical; the simultaneous control over all subconfigurations, plus the matching negative example, is the actual contribution. The statements are precise, the hypotheses (linear independence over Z, vanishing constants) are stated up front, and the counterexample is calibrated against the natural 1/8 benchmark. Within additive combinatorics and finite-field ergodic theory this is the right next question, and they answer both the positive and negative sides cleanly.\n\nThe soft spots are proportional to the fact that we only have the abstract. The counting or Fourier/Gowers arguments that produce the common d cannot be checked, nor can the explicit construction of the density-1/2 set. Linear independence and zero constants are load-bearing—exactly as the paper's own negative result illustrates—so the claim is tightly conditioned, but that is not a hidden flaw; it is the reason the hypotheses appear. Quantitative dependence on p and the polynomials is also invisible here.\n\nThis is for people who already care about popular differences, polynomial Szemerédi, or Gowers norms over finite fields. A serious referee should see the full proofs; the results are important enough inside the subfield and look carefully formulated. I would send it out for review rather than desk-reject.","headline":"Clean simultaneous upgrade of Green's popular-difference theorem to polynomials over large F_p, plus a sharp high-dimensional counterexample; abstract-only so proofs unchecked.","tokens_in":3254,"tokens_out":498,"would_cite":false,"duration_ms":11437,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11B30","11T06","05D10"],"pacs":[],"model":"grok-4.5","headline":"Linearly independent polynomials over finite fields share a single popular difference that works for every subconfiguration at once.","keywords":["popular differences","polynomial configurations","finite fields","simultaneous popular differences","arithmetic progressions","Green's theorem","linear independence"],"falsifier":"Exhibit a fixed linearly independent zero-constant family for which, in arbitrarily large prime fields, every dense set admits some nonzero d that fails the random-set lower bound for at least one subconfiguration by a fixed positive amount.","tokens_in":3231,"feed_emoji":"∑","tokens_out":602,"duration_ms":4643,"temperature":0.7,"pith_summary":"Green's popular difference theorem guarantees a single nonzero d that makes the three-term arithmetic progression almost as common as random chance would predict inside any dense set over a large prime field. This paper strengthens that guarantee to whole families of polynomial configurations: for any fixed collection of linearly independent polynomials with zero constant terms, a single d works simultaneously for every subproduct of the corresponding indicators. The density lower bound is the natural random-set value for each subconfiguration, up to an arbitrarily small error. The same paper shows that the simultaneous phenomenon cannot be taken for granted: over vector spaces F_p^n with p fixed and n large, no such d can make both the ordinary progression and the progression with common difference 2d popular at the same time, for a set of density roughly one half. The positive result therefore relies on the polynomial family being independent and on the ambient field being a large prime field rather than a high-dimensional vector space over a fixed field.","feed_headline":"One d works for every polynomial subconfiguration at once","feed_subtitle":"Linearly independent zero-constant polynomials share a popular difference; the simultaneous guarantee fails in high dimension","key_machinery":"The simultaneous popular-difference statement for a linearly independent family of zero-constant-term polynomials: a single nonzero d that realises the random-set lower bound for every 0-1 weighting of the product of indicator functions.","core_discovery":"For every fixed collection of linearly independent integer polynomials with zero constant terms, every ε>0, all large enough primes p, and every set A of density α in F_p, there exists a single nonzero d such that the density of every subconfiguration formed by a subset of those polynomials is at least α to the power of one plus the number of polynomials used, minus ε. The same simultaneous guarantee fails for the pair of three-term progressions with differences d and 2d over F_p^n.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["One d serves every polynomial subconfiguration at once over F_p","Shared popular d for all subsets of independent polynomials","Single nonzero d lifts every subconfig density near α power","Polynomial configs share one popular difference simultaneously","Simultaneous popular diffs hold for zero-constant polynomials"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The polynomials must be linearly independent over the integers and must all have zero constant term; without those structural hypotheses the existence of a single popular d for every subconfiguration is not claimed.","fun_headline_variants_meta":{"raw":{"variants":["One d serves every polynomial subconfiguration at once over F_p","Shared popular d for all subsets of independent polynomials","Single nonzero d lifts every subconfig density near α power","Polynomial configs share one popular difference simultaneously","Simultaneous popular diffs hold for zero-constant polynomials"]},"model":"grok-4.5","effort":"low","cost_usd":0.004568,"raw_usage":{"total_tokens":1564,"prompt_tokens":1108,"num_sources_used":0,"completion_tokens":78,"cost_in_usd_ticks":45680000,"prompt_tokens_details":{"text_tokens":1108,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":378,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":1108,"tokens_out":78,"duration_ms":3219,"temperature":1.0,"reasoning_tokens":378,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T00:45:00.528380+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a fixed linearly independent zero-constant family for which, in arbitrarily large prime fields, every dense set admits some nonzero d that fails the random-set lower bound for at least one subconfiguration by a fixed positive amount.","supporting_citations":[],"review_version":1}