{"id":"2fa951b9-f289-47e6-ad8b-9ee34f6e3734","arxiv_id":"2506.13010","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For any integer polynomial P with P(0)=0, any subset of [N] avoiding x, x+P(y), ..., x+kP(y) has size at most N (log log log N)^{-c}, with stronger bounds when P'(0)!=0.","lead":"The paper proves new effective bounds for polynomial Szemerédi-type progressions over integers and finite fields, including the first reasonable bounds for progressions with a fixed polynomial common difference. A reader interested in additive combinatorics will find a general transference method based on nilsequence comparison rather than degree-lowering.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 2.9 Step 2 asserts that two oscillatory integrals are identical after a change of variables, but the identity is not derived; if the scaling is off by a factor of z, W, or N^{1/2}, the claimed N^{-Ω(1)} comparison fails and the transference argument for Theorem 1.1 collapses.","rationale":"The reader's weakest-assumption pick, Lemma 2.9, is correct. My stress-test narrows the concern to a single unverified algebraic identity inside Step 2, rather than the whole lemma or the imported equidistribution theorems. This is load-bearing because Lemma 2.9 is the only place where the original polynomial pattern is quantitatively compared with the model pattern; without a difference bound strong enough to contradict the quasipolynomial inverse theorem, the stashing argument in Section 2.4 cannot go through. This is not a circularity or an external-consensus issue: it is a concrete, checkable step. The Hensel-type lemmas (2.1, 2.2) are complete, the finite-field transference in Section 3 is written in more detail, and the reliance on [28] and [24] is external but not circular. Nothing in the manuscript suggests bad faith; the issue is a verifiable technical gap. The proposed symbolic substitution and numerical sanity check are cheap and would definitively settle whether the cancellation holds. If the identity holds, the remaining omissions in Lemma 2.9 are presentation gaps and the paper merits ACCEPT after filling them; if it fails, the proof of Theorem 1.1 needs repair. Since the reader already recommended CONDITIONAL, my recommendation leaves the verdict unchanged.","tokens_in":35242,"tokens_out":10404,"duration_ms":106998,"concrete_test":"Reproduce the final change-of-variables step in the q₂ = 1 case of Lemma 2.9 with a concrete example, e.g. P(y) = y² − y⁴ (so d = 4, d′ = 2, ε = −1, b_d = −1) and Θ′ = 1/N. Substitute y = (zN/|b_d|W^{d-d′})^{1/d} u in the first integral and the corresponding substitution in the second integral, and check whether the two integrands and limits are exactly equal after inserting the definition of ν(y). If the phases differ by a factor such as N^{1/2} z W or a power of z, the claimed N^{-Ω(1)} bound is invalid. A numerical evaluation of both sides for several large N and z ∈ [1/2,1] would settle the identity.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.1 rests on the transference setup in Section 2, whose quantitative core is Lemma 2.9: the comparison of nilsequence averages over P_W(y) and Q(z,y) = (ε z W + 1)y^{d'}. After Step 1 reduces the problem to linear exponential phases, Step 2 handles the approximating phase Θ = a/q + Θ' with q ≤ W^{O(1)} and |Θ'| ≤ W^{O(1)}/N. In the case q₂ = 1 (all prime factors of q at most w), the proof asserts the stronger bound ≤ N^{-Ω(1)}. The decisive displayed step converts the two Riemann sums into integrals and claims that, after substituting for y and using the definition of ν(y), the two integrals are 'intentionally' identical. This equality is not demonstrated; it is a genuine check because the first integrand has phase Θ' b_d W^{d-d'} y^d with range (zN/(|b_d|W^{d-d'}))^{1/d}, while the second has phase Θ' N^{1/2} ε z W y^{d'} with range (N/W²)^{1/(2d')}, and the change of variables must exactly absorb the factors N^{1/2}, z, W, and the weight ν(y). A silent scaling error here would leave the difference not O(N^{-Ω(1)}), which is exactly the error budget needed for the contradiction with the quasipolynomial inverse theorem in Section 2.4. The manuscript itself warns that details are omitted ('we omit the details' in Lemma 2.7; 'we only sketch it' for the second case in Step 1 of Lemma 2.9), and this integral identity is the most concrete point where an unchecked step would propagate directly to Theorem 1.1.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves new quantitative bounds for the polynomial Szemerédi theorem. Theorem 1.1 gives, for fixed P(y) in Z[y] with P(0)=0 and distinct integers a_1,...,a_t, bounds of the form N exp(-c(log log N)^c) for t=3 and P'(0) not equal to 0, N exp(-c(log log log N)^c) for all t with P'(0) not equal to 0, and N (log log log N)^{-c} for P'(0)=0, for sets A subset [N] avoiding x, x+a_2 P(y), ..., x+a_t P(y). The proof passes through a W-tricked polynomial P_W(y), compares the counting operator Lambda_W with a model operator Lambda_Model via stashing and an inverse theorem for Gowers norms, and reduces the core comparison to a nilsequence equidistribution statement (Lemma 2.9). Over Z/NZ, Theorem 1.6 proves a transference statement for 'transferable' polynomial patterns, yielding r_P(Z/NZ) bounded by N exp(-c'(log log log N)^{c'}), and the paper introduces the notion of transferable polynomial patterns (Definition 1.4).","tokens_in":35607,"tokens_out":13260,"duration_ms":132555,"significance":"If fully substantiated, Theorem 1.1 is the first reasonable quantitative bound for Szemerédi's theorem with an arbitrary fixed polynomial common difference and arbitrary length, and Theorem 1.6 unifies and extends several known finite-field cases, including results conjectured by Leng. The transference framework using stashing and the algebraic reduction to nilsequence comparison is a substantial structural contribution, and the notion of transferable polynomial patterns is natural and potentially reusable. The authors are explicit that several key estimates are sketched or deferred: Lemma 2.7 ends with 'we omit the details', Lemma 2.9's second case is described as 'we only sketch it', and the deduction of Theorem 1.6 from Theorem 3.1 is omitted. These are load-bearing gaps, so the significance is conditional on completing them.","major_comments":[{"comment":"The proof of the key convolution estimate (2.6) stops at 'the desired result follows via direct integration on the sizes of level sets of |F(Theta)|; we omit the details.' This is load-bearing: the entire reduction of Lambda_Model to Gowers-Peluse norm control in Lemma 2.7 depends on this claim, and the final polynomial dependence on delta is determined by the choice of m and the level-set measure. Please supply the missing integration argument, including the precise threshold on eta implied by the condition eta^{-O(1)} <= N^{1/2}/W, and state the resulting quantitative dependence on delta.","section":"Lemma 2.7, proof of (2.6)"},{"comment":"The final displayed comparison for the q_2 = 1 case asserts that the two integrals are 'intentionally' identical after a change of variables, but the calculation is not shown. The change of variables must simultaneously absorb N^{1/2}, z, W, and the weight nu(y). A direct calculation with u = N^{1/2} epsilon z W y^{d'} does appear to confirm the identity for epsilon = +1, but the case b_d < 0 and the conversion from Riemann sums to integrals (with its error term) are not discussed. Since this identity is where the N^{-Omega(1)} error budget is spent, please include the full calculation and justify all error terms.","section":"Lemma 2.9, Step 2"},{"comment":"The treatment of the model-operator average is introduced with 'we only sketch it'. This is the half of the inductive step that produces the characters eta_j and the uniform derivative bound (2.12) for the Q(z,y) average, and it is used in every application of Lemma 2.9, hence in the proof of Theorem 1.1. Please expand this sketch into a complete argument, in particular the downward induction via the leading coefficient z times the binomial coefficient (y choose k), and the treatment of the nu(y) weight in the averaging argument.","section":"Lemma 2.9, Step 1, second case"},{"comment":"The proof asserts that transferability implies V = Psi[1] x ... x Psi[k] 'after a moment's thought' and that the denominators introduced when writing (0, ..., v_{i,j,ell}, ..., 0) as a Q-linear combination of the (v_{i,j,1}, ..., v_{i,j,k}) depend only on P and k. This identification is the algebraic heart of Theorem 3.4 and, through Theorem 3.1, of Theorem 1.6. Please provide a detailed linear-algebra proof of (3.4) and of the uniform denominator bound, since the current presentation leaves the main algebraic step to the reader.","section":"Theorem 3.4, Eq. (3.4)"}],"minor_comments":[{"comment":"The abstract states only the N (log log log N)^{-Omega(1)} bound, while Theorem 1.1 gives the stronger N exp(-c(log log N)^c) for t=3 with P'(0) not equal to 0 and N exp(-c(log log log N)^c) for all t with P'(0) not equal to 0. Please align the abstract with the full theorem statement.","section":"Abstract vs. Theorem 1.1"},{"comment":"In the case (p,d!) not equal to 1, the proof says 'we may assume without loss of generality that d < k' before writing [p^k] = p^d [p^{k-d}] + [p^d]; this decomposition requires k >= d, and the case k <= d should be handled separately. Also the lemma states c_j congruent to 0 mod p^{2d} for j >= r+1, while the proof in that case only appears to use c_j congruent to 0 mod p^d; please reconcile the statement with the proof.","section":"Lemma 2.2, proof"},{"comment":"The two Manners references appear to refer to the same work: the text cites [33] for 'stashing' after citing [32] for the same concept. Please consolidate the references and choose one citation for the stashing argument.","section":"References [32] and [33]"},{"comment":"The derivation of Theorem 1.6 from Theorem 3.1 is explicitly omitted ('We omit the details of this deduction because they are essentially the same as in the previous section'). Since Theorem 1.6 is a headline result, please include the supersaturation deduction or state precisely which supersaturation results are being used and how they are applied.","section":"Section 3, deduction of Theorem 1.6"},{"comment":"Definition 1.4 uses kappa(P) from Definition 1.3, but Definition 1.3 is stated for polynomials in several variables while Definition 1.4 treats P = (x + P_1(y), ..., x + P_t(y)). Please make the notational correspondence explicit so that the kernel system is understood as taken over the one-variable polynomials P_i(y).","section":"Definition 1.4"}],"recommendation":"major_revision","confidential_remarks":"The paper depends heavily on unpublished work by the same authors or close collaborators, including [27], [28], [39], [1], [24], and [17]. This is not itself an error, but it makes independent verification difficult, and the editor may wish to ask the authors to state explicitly which theorems are imported and whether each import is available in final form. The paper does not appear to be internally circular or to fit parameters to conclusions, but the omitted details in Lemmas 2.7 and 2.9 are substantial enough that I could not certify the proof in its present form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: the main integer theorem (Theorem 1.1) is not proven, because the key nilsequence comparison Lemma 2.9 contains a false identity. The finite-field side (Theorem 1.6) and the transferable-pattern framework are genuinely interesting and may well be correct.\n\nWhat's new: the paper defines transferable patterns and gives a unified transference result over Z/NZ that covers several prior cases and some new ones. The proof strategy—iterate stashing, reduce to nilsequences, then compare orbits on nilmanifolds—is a substantial new idea. If the integer part could be repaired, Theorem 1.1 would be a major advance, since earlier effective results only handled two-term patterns, differing-degree patterns, or special common differences.\n\nThe problem: in Step 2 of Lemma 2.9, after reducing to linear exponential phases and q2=1, the proof converts two Riemann sums into integrals and claims that after a change of variables the two integrals are 'intentionally' identical. I checked the algebra. With the weight nu as defined in Definition 2.5, the first term becomes (Nz)^{-1/d} (d/d') ∫ u^{d/d'-1} e(Theta' epsilon u^{d'}) du and the second becomes (Nz)^{-1/d} (d'/d) ∫ y^{-(d-d')/d} e(Theta' epsilon y^{d'}) dy. These are not equal; the prefactors are reciprocals and the exponents only match when d=d'. So the claimed N^{-Omega(1)} cancellation is not established. This is the exact step that supplies the contradiction in the stashing argument, so Theorem 1.1 collapses at this point.\n\nTwo other things worth noting. Lemma 2.7 ends with 'we omit the details' for the exponential-sum claim (2.6); that's a minor gap compared with the integral identity, but it's another place the reader has to fill in. And the proof leans on several unpublished preprints by the same group; that raises the verification burden but is not a flaw in itself.\n\nThe finite-field proof in Section 3 is written at a somewhat higher level of care and the algebraic comparison in Theorem 3.4 looks plausible, so I would not dismiss the whole paper.\n\nWho should read it: anyone working on quantitative polynomial Szemeredi theorems. My recommendation: send it to referees, but clearly flag the Lemma 2.9 integral identity. The authors need to either fix the weight/range choice or produce a different comparison argument.","headline":"Theorem 1.1 has a concrete error in Lemma 2.9's integral comparison, so the main integer result does not go through as written; the finite-field framework is still promising.","tokens_in":36160,"tokens_out":7016,"would_cite":false,"duration_ms":65500,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11B30","11L07"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any fixed polynomial P with P(0)=0, sets in [N] avoiding x, x+a2P(y), ..., x+atP(y) are smaller than N divided by a triply logarithmic factor, with stronger bounds in special cases.","keywords":["polynomial Szemerédi theorem","transference","Gowers norms","Gowers-Peluse norms","nilsequences","supersaturation","W-trick","quantitative density bounds"],"falsifier":"Compute the two averages in Lemma 2.9 for $P(y)=y^2-y^4$, with $W$ chosen as in (2.1) for $w=(\\log N)^{1/2}$, and with a degree-2 nilsequence whose horizontal and quadratic frequencies have denominators between $w$ and $W^{O(1)}$. If for any admissible residue class the difference exceeds $\\exp(-c(\\log\\log N)^c)$ at the stated ranges of $N$ and $W$, the central transference bound is false; a cheaper diagnostic is whether the incomplete exponential-sum estimate at the end of Step 2 remains valid when the rational denominator is exactly $W^{O(1)}$ and the residual phase is $O(W/N)$.","tokens_in":34995,"feed_emoji":"🔢","tokens_out":13319,"duration_ms":139233,"temperature":0.7,"pith_summary":"The paper proves quantitative bounds for the polynomial Szemerédi theorem when the common difference is a fixed polynomial. Specifically, for any $P(y)\\in\\mathbb{Z}[y]$ with $P(0)=0$ and any distinct integers $a_1,\\ldots,a_t$, every subset $A\\subseteq[N]$ with no configuration $x, x+a_2P(y),\\ldots,x+a_tP(y)$ has size at most $N\\exp(-c(\\log\\log N)^c)$ when $t=3$ and $P'(0)\\ne 0$, at most $N\\exp(-c(\\log\\log\\log N)^c)$ when $P'(0)\\ne 0$, and at most $N(\\log\\log\\log N)^{-c}$ when $P'(0)=0$, with $c=c(\\deg P,t)>0$. The interest is that previous effective results covered only special patterns such as two-term polynomial differences, homogeneous perfect-power differences, or progressions whose polynomials have distinct degrees. A sympathetic reading is that the paper establishes the first reasonable quantitative version of Szemerédi's theorem with an arbitrary fixed polynomial common difference and arbitrary length.","feed_headline":"Triple-log bounds now cover every fixed polynomial progression","feed_subtitle":"The density of sets avoiding a fixed polynomial progression decays at least like a triple-log factor","key_machinery":"The load-bearing object is the pair of counting operators $\\Lambda_W$ and $\\Lambda_{\\mathrm{Model}}$ together with the algebraic notion of a transferable polynomial pattern. A pattern is transferable if every polynomial relation in its kernel system is already a relation among its homogeneous linearized parts $P_i^*(y_1,\\ldots,y_d)$; this is exactly the condition needed for the nonlinear and linearized orbits on a nilmanifold to equidistribute on the same subnilmanifold. The analytic engine is an iterative Cauchy–Schwarz stashing argument: starting from a large difference between $\\Lambda_W$ and $\\Lambda_{\\mathrm{Model}}$, each dual function is shown to have large Gowers or Gowers–Peluse norm, the quasipolynomial inverse theorem converts this into a nilsequence, and the problem becomes a comparison of two polynomial orbits on a nilmanifold. That comparison is carried out by an iterative step-down reduction (Lemma 2.9) using quantitative equidistribution results and Hensel-type lemmas showing that $P_W(y)$ and $(\\varepsilon zW+1)y^{d'}$ have identical distributions modulo $W^{d-d'}$.","core_discovery":"The central discovery is that the nonlinear pattern $x, x+a_2P(y),\\ldots,x+a_tP(y)$ can be quantitatively transferred to a linear model pattern without any degree-lowering hypothesis. The paper proves that the count $\\Lambda_W(f_1,\\ldots,f_t)$ of the pattern with W-normalized difference $P_W(y)=W^{-d'}P(Wy)$ differs from the model count $\\Lambda_{\\mathrm{Model}}(f_1,\\ldots,f_t)$ with difference $(\\varepsilon zW+1)y^{d'}$ by at most $\\exp(-c(\\log\\log N)^c)$ for 1-bounded functions supported on $[N]$. Once this transference is available, density bounds follow from known supersaturation statements for arithmetic progressions with shifted square differences. Over $\\mathbb{Z}/N\\mathbb{Z}$ with $N$ prime, the analogous transference statement holds for every transferable polynomial pattern, with error $\\exp(-c(\\log\\log N)^c)$, and yields density bounds $N\\exp(-c'(\\log\\log\\log N)^{c'})$.","pith_inferences":["The author leaves implicit that the transference error is likely self-improving: if the supersaturation input in Lemma 2.11 is improved to the conjecturally optimal arithmetic-progression bounds, the corresponding case of Theorem 1.1 improves without any change in the stashing scheme.","A natural extension is to test whether transferability is necessary as well as sufficient; the paper's non-homogeneous example suggests that non-transferable patterns may require genuinely new ideas beyond a better equidistribution theorem.","A concrete numerical check of Lemma 2.9 for $P(y)=y^2-y^4$ on a step-2 nilmanifold would reveal whether the $P'(0)=0$ bound is an artifact of the proof or reflects the true difficulty of the sign conflict between archimedean and p-adic behavior."],"forward_implications":["For every fixed $P$ with $P(0)=0$, the maximum size of a set avoiding $x, x+a_2P(y),\\ldots, x+a_tP(y)$ is at most $N\\exp(-c(\\log\\log\\log N)^c)$, and at most $N\\exp(-c(\\log\\log N)^c)$ when $t=3$ and $P'(0)\\ne 0$.","The finite-field polynomial Szemerédi theorem holds with quantitative decay $N\\exp(-c(\\log\\log\\log N)^c)$ for every transferable polynomial pattern, covering the earlier effective finite-field cases in one framework.","The transference error between the original polynomial pattern and its linearized model is quantitatively negligible, so future improvements in supersaturation bounds for linear patterns automatically transfer to improved integer density bounds.","The W-trick and archimedean weights lose only subpolynomial factors in $N$, keeping the bounds in the reasonable regime rather than the tower-type regime typical of density-increment arguments."],"supporting_citations":[{"why":"Supplies the Gowers-norm control for polynomial operators over the integers (Theorem B.3) that starts the stashing argument.","marker":"[36]"},{"why":"Supplies the quasipolynomial inverse theorem for the Gowers norm that turns large $U^s$ norms into nilsequences.","marker":"[28]"},{"why":"Supplies the imported quantitative equidistribution theorem (Theorem 2.8) used in the step-down nilsequence comparison.","marker":"[24]"},{"why":"Supplies the supersaturation bound for homogeneous polynomial patterns used to obtain positivity of the model operator when $d'>1$.","marker":"[40]"},{"why":"Supplies the earlier transference framework with archimedean weights and the W-trick that the integer argument adapts.","marker":"[39]"},{"why":"Supplies the quantitative theory of polynomial orbits on nilmanifolds used in the finite-field equidistribution argument.","marker":"[14]"},{"why":"Supplies the Lie-algebra perspective on transferable polynomial patterns used in the proof of Theorem 3.4.","marker":"[1]"}],"fun_headline_variants":["Triple-log decay for every fixed polynomial pattern","Transference nails polynomial Szemeredi with triple-log bounds","All fixed polynomial progressions now get triple-log density limits","Reasonable bounds for polynomial progressions: triple-log achieved","Nonlinear progressions: triple-log bounds via quantitative transference"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole argument assumes that, after the W-trick removes small-prime biases and a weight corrects for density, the polynomial $P(y)$ and the shifted pure-power difference $(\\varepsilon zW+1)y^{d'}$ behave identically for every residue class; if that comparison fails, the transference step collapses.","fun_headline_variants_meta":{"raw":{"variants":["Triple-log decay for every fixed polynomial pattern","Transference nails polynomial Szemeredi with triple-log bounds","All fixed polynomial progressions now get triple-log density limits","Reasonable bounds for polynomial progressions: triple-log achieved","Nonlinear progressions: triple-log bounds via quantitative transference"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000326,"raw_usage":{"total_tokens":1788,"prompt_tokens":871,"completion_tokens":917,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":487,"completion_tokens_details":{"reasoning_tokens":836}},"tokens_in":487,"tokens_out":917,"duration_ms":10819,"temperature":1.0,"reasoning_tokens":836,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T00:38:21.544336+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the two averages in Lemma 2.9 for $P(y)=y^2-y^4$, with $W$ chosen as in (2.1) for $w=(\\log N)^{1/2}$, and with a degree-2 nilsequence whose horizontal and quadratic frequencies have denominators between $w$ and $W^{O(1)}$. If for any admissible residue class the difference exceeds $\\exp(-c(\\log\\log N)^c)$ at the stated ranges of $N$ and $W$, the central transference bound is false; a cheaper diagnostic is whether the incomplete exponential-sum estimate at the end of Step 2 remains valid when the rational denominator is exactly $W^{O(1)}$ and the residual phase is $O(W/N)$.","supporting_citations":[{"cited_title":"Pi8 (2020), e16, 55","cited_arxiv_id":null,"evidence_quote":"Supplies the Gowers-norm control for polynomial operators over the integers (Theorem B.3) that starts the stashing argument."},{"cited_title":"Efficient Equidistribution of Nilsequences","cited_arxiv_id":"2312.10772","evidence_quote":"Supplies the imported quantitative equidistribution theorem (Theorem 2.8) used in the step-down nilsequence comparison."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the supersaturation bound for homogeneous polynomial patterns used to obtain positivity of the model operator when $d'>1$."},{"cited_title":"Effective bounds for Roth's theorem with shifted square common difference","cited_arxiv_id":"2309.08359","evidence_quote":"Supplies the earlier transference framework with archimedean weights and the W-trick that the integer argument adapts."},{"cited_title":"Second Series175 (2012), 465–540","cited_arxiv_id":null,"evidence_quote":"Supplies the quantitative theory of polynomial orbits on nilmanifolds used in the finite-field equidistribution argument."},{"cited_title":"A non-flag arithmetic regularity lemma and counting lemma","cited_arxiv_id":"2209.14083","evidence_quote":"Supplies the Lie-algebra perspective on transferable polynomial patterns used in the proof of Theorem 3.4."}],"review_version":1}