{"id":"3dae4e44-7b7e-432f-b000-a470058c120a","arxiv_id":"2507.14414","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Weighted multilinear polynomial averages in finite fields are controlled by the u^(d+1)-norm of the weight, yielding quantitative convergence and new asymptotic formulas for multidimensional rational function progressions of arbitrary length.","lead":"Weighted averages of products f_i(x + P_i(y)v_i) over finite fields are shown to converge, at a polynomial rate in the prime p, to averages over independent variables whenever the weight is strongly uniform in a u^(d+1)-norm sense. This yields the first asymptotic formulas for multidimensional rational function progressions of arbitrary length, previously known only for three terms.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 1.4 skips the uniform Bombieri estimate: the pushforward measure under φ must be shown strongly (d+1)-uniform, i.e. all rational-phase sums E^*_y ep(Q(φ(y))) for deg Q ≤ d must decay uniformly; this is asserted, not proved.","rationale":"The main theorem (Theorem 1.2) is supported by a coherent degree-lowering induction, and the box-norm control in Section 3 is structurally sound. The dual-function calculations omit complex conjugates throughout Section 3 (e.g. the definitions of F^θ_{l,k} and the box norm use products without conjugation), which makes the displayed identities false for complex-valued functions as written. However, the argument is valid for real-valued functions, and any 1-bounded complex function decomposes into real and imaginary parts, so this is a presentation/technical gap rather than a fatal flaw in the central theorem. The genuinely load-bearing soft spot is Proposition 1.4, exactly as the reader identified. The passage from the polynomial weighted theorem to the rational-function progression requires viewing E^*_y as the pushforward measure ν under φ and then verifying that ν is strongly (d+1)-uniform. That verification is a uniform family of exponential sum estimates E^*_y ep(Q(φ(y))) for all Q of degree at most d, with constants independent of Q's coefficients. Bombieri's bound applied only to a single phase does not automatically give the uniform sup over all Q, and the paper does not supply the argument. The uniformity is in fact true for nonconstant φ because deg(Q∘φ) is bounded by d·deg φ < p for large p, which rules out the only exceptional case in the Weil bound (the Frobenius-type phases A^p-A+c); but this is a nontrivial verification that must be written down for Proposition 1.4 to be considered proved. The additional observation that 'nonzero' must mean 'nonconstant' is a small but important correction: a nonzero constant φ makes the proposition false. I therefore support the reader's CONDITIONAL verdict; the identified concerns are local, fixable, and do not undermine the main polynomial theorem, but they do need to be addressed before the rational-function application is accepted as proved.","tokens_in":9995,"tokens_out":26773,"duration_ms":228004,"concrete_test":"Write out the missing uniformity lemma: for fixed nonconstant φ and d, prove sup_{Q∈Z[y], 1≤deg Q≤d} |E^*_y ep(Q(φ(y)))| = O_{d,φ}(p^{-1/2}), tracking Bombieri's exceptional-phase condition and checking that Q∘φ cannot be a Frobenius-type phase when p > d·deg φ. Then re-derive Proposition 1.4 by applying Theorem 1.3 to the pushforward measure ν. As a numerical spot-check, for φ(y)=1/y and d=2, compute max_{a,b,c∈F_p, (a,b,c)≠0} |E^*_{y∈F_p} ep(a/y^2+b/y+c)| for primes p∈{1009,2003,4001} and confirm the maximum is ≤ C/√p with C independent of p; also test the k=1 instance of Proposition 1.4 with random 1-bounded f_0,f_1 and confirm the error decays.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central new application is Proposition 1.4: for a rational function φ, the average over y of f_i(x+P_i(φ(y))v_i) is claimed to be close to the linear average. The only route from Theorem 1.3 is to write the y-average as a weighted polynomial average with weight ν(z) equal to the pushforward of the uniform measure on F_p under φ, so that E^*_y F(φ(y)) = E_z ν(z) F(z). Theorem 1.3 then applies only if ν is strongly (d+1)-uniform. This condition is exactly: for every nonconstant Q ∈ Z[y] with deg Q ≤ d, |E^*_y ep(Q(φ(y)))| ≤ C p^{-1/2}, with C independent of the coefficients of Q, and with the same power saving for all Q simultaneously. The paper states 'as a direct application of Theorem 1.3 and Bombieri's bound' (Section 1.3) but never defines ν, never states this uniformity lemma, and never rules out the exceptional phases in Bombieri's bound (phases of the form A(y)^p - A(y) + c) after composition Q∘φ. Without uniform control, the O(p^{-c}) error in Theorem 1.3 cannot be summed over the p^{Dk} Fourier coefficients, so Proposition 1.4 is not established. Also, the hypothesis 'nonzero rational function' must be read as 'nonconstant': if φ is constant, Proposition 1.4 is false, e.g. k=1, φ≡1, P_1(y)=y, v_1=1, where the left side is E_x f_0(x)f_1(x+1) but the right side is E_x f_0(x) E_n f_1(x+n). These are fixable: the uniformity follows from Weil/Bombieri because deg(Q∘φ) < p for large p, ruling out Frobenius degeneracy, and the constant case is excluded by adding 'nonconstant'. But as written, the load-bearing step for the rational-function application is missing.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies weighted multilinear polynomial averages over F_p, proving a quantitative L^2 convergence result (Theorem 1.2) under the assumption that the weight θ is 1-bounded and strongly (d+1)-uniform with respect to the u^s-norm introduced in Definition 1.1. The proof is an induction on the number of polynomial factors, combining a box-norm control lemma (Lemma 3.3) with a degree-lowering step that reduces the multilinear average to the u^{d+1}-norm of θ; the unweighted multidimensional polynomial Szemerédi theorem of Kuca is used as a black box. The paper further claims a rational-function application (Proposition 1.4) for progressions x, x+P_1(φ(y))v_1, ..., x+P_k(φ(y))v_k, stated as a direct consequence of Theorem 1.3 and Bombieri's exponential-sum bound, together with a density corollary (Corollary 1.5). The main technical core of Section 3 is coherent given Kuca's theorem, but the complex-valued formulation is not justified as written, and the rational-function application contains a substantial gap and a literal counterexample for constant φ.","tokens_in":10296,"tokens_out":16461,"duration_ms":181168,"significance":"If the identified issues are repaired, the paper would provide a clean quantitative criterion for L^2 convergence of weighted multilinear polynomial averages in finite fields, with an error term that is power-saving in p. The u^s-norm formulation, inspired by Teräväinen, is a useful simplification of earlier Gowers-norm approaches, and the induction in Section 3 is a coherent extension of Peluse--Kuca methods to weighted averages. The rational-function application, if proved, would go beyond the three-term rational-function results currently in the literature and would be a notable contribution. The reliance on Kuca's theorem as a black box is appropriate, and Lemma 2.2 is proved in the text. However, the missing conjugation in the complex-valued argument and the unproved uniformity in Proposition 1.4 are load-bearing issues that must be addressed before the claims are established.","major_comments":[{"comment":"The argument is written as if all functions are real-valued, but Theorems 1.2 and 1.3 are stated for 1-bounded complex-valued functions. The definition of the multiplicative derivative, △_h f(x) = f(x)f(x+h), the identity in Lemma 2.2, and the first displayed identity in the proof of Lemma 3.3, ||G^θ_{l,k}||^2_{L^2} = E_x F^θ_{l,k}(x) f_l(x), all need complex conjugates on alternating factors (or on θ(y') and f_i(x+P_i(y')v_i)) to be correct for complex-valued f. Likewise, the inner products ⟨f,g⟩ in the proof of Theorem 1.2 are written as bilinear forms without conjugation. No reduction to real and imaginary parts is stated. Since these identities drive the induction, the complex-valued theorem is not established by the text as written. This is repairable: one can insert the missing conjugates throughout, or prove the real-valued case and pass to complex functions by decomposing each f_i into real and imaginary parts.","section":"Section 3, Lemmas 2.2 and 3.3"},{"comment":"The claim that Proposition 1.4 follows 'as a direct application of Theorem 1.3 and Bombieri's bound' is not justified. To apply Theorem 1.3, one must introduce the pushforward measure ν(z) = E^*_y 1_{φ(y)=z} on F_p and prove that ν is strongly (d+1)-uniform in the sense of Definition 1.1. This requires a uniform estimate |E^*_y ep(Q(φ(y)))| ≤ C p^{-c} for every polynomial Q of degree at most d, with C and c independent of Q and p. Bombieri's bound is normally stated for a fixed rational phase and has exceptional phases; the paper neither states a uniform version for the family Q∘φ nor rules out exceptional behavior after reducing the coefficients. Without such a uniform bound, the passage from Theorem 1.3 to Proposition 1.4 is not established, and consequently Corollary 1.5 is not proved either.","section":"Section 1.3, Proposition 1.4"},{"comment":"The hypothesis 'nonzero rational function' is insufficient as stated. If φ(y) ≡ 1, k = 1, P_1(y) = y, and v_1 = 1, then the left-hand side of Proposition 1.4 is E_x f_0(x) f_1(x+1), while the right-hand side is E_x f_0(x) E_n f_1(x+n). These are not equal for general 1-bounded functions, e.g. f_1 = 1_{\\{0\\}}. The proposition becomes plausible only after adding the assumption that φ is nonconstant (and after resolving the uniformity issue raised above). This should be fixed explicitly.","section":"Section 1.3, Proposition 1.4"}],"minor_comments":[{"comment":"The operator GEθ_k is used in equation (4) but is never defined; it appears to denote Eθ times the unweighted operator G^1_k, which is distinct from GEθ. Please define it explicitly to avoid confusion.","section":"Section 3, proof of Theorem 1.2"},{"comment":"The Gowers box norm should be defined with complex conjugates on the appropriate factors so that the expression is real and nonnegative; as written, for complex-valued f the quantity need not be a norm.","section":"Section 2.2, Definition 2.2"},{"comment":"The exponent O_P(1) in the L^2-norm estimate is left unspecified; stating a concrete exponent (for instance, 16) would make the induction and the subsequent Cauchy-Schwarz steps easier to verify.","section":"Section 3, Proposition 3.2"},{"comment":"There is a typo in the set notation '{P1, · · ·, Pk ∈ Z[y}' where a closing brace is missing; it should be '\\{P_1, \\dots, P_k\\} \\subset \\mathbb{Z}[y]'.","section":"Section 1.2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the scope of the journal and the main technical idea is promising. The two load-bearing concerns are the missing conjugation in the complex-valued argument of Section 3 and the unproved uniform Bombieri bound needed for Proposition 1.4. Both appear fixable. The constant-φ counterexample is easily handled by adding 'nonconstant.' No concerns about citation practices; the self-citation [HL25] appears only as background context."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know up front. The core result is real: Proposition 3.2 gives u^{d+1}-norm control for weighted polynomial averages in F_p^D, and Theorem 1.2 converts that into quantitative L2 convergence for strongly uniform weights. I read the induction; it is coherent, and the use of Kuca's theorem as a black box is legitimate. But the advertised rational-function application, Proposition 1.4, is not established by the text, and the statement is false as written for constant φ. Take φ≡1, k=1, P_1(y)=y, v_1=1: the left side is E_x f0(x)f1(x+1), while the right side is E_x f0(x) E_n f1(x+n). Those differ.\n\nWhat is actually new: the multidimensional u^{d+1}-norm control is new for D≥2, extending Teräväinen's D=1. The proof adapts Kuca's box-norm framework without hiding anything significant; Lemma 2.2 is proved in the text, and the degree-lowering induction checks out. There is no circularity; the main theorem rests on Kuca's unweighted theorem and a classical inverse box-norm bound.\n\nThe soft spots are real but most are fixable. First, the paper declares complex-valued functions but defines the box norm, dual functions, and inner products without complex conjugation. The computations are valid for real-valued functions; no reduction is stated. This is a standard fix, but it needs to be said. Second, Proposition 1.4's \"direct application\" skips the actual mechanism: one must define the pushforward measure ν(z)=E^*_y 1_{φ(y)=z}, check it is O_φ(1)-bounded, and prove it is strongly (d+1)-uniform. That uniformity is a uniform Bombieri bound over all Q of degree ≤ d. For fixed nonconstant φ and p larger than the degrees of Q∘φ, the exceptional A^p−A+c case has degree ≥ p and so is ruled out; the bound follows. But the text does none of this, and it also does not state that φ must be nonconstant. The constant case is not a corner case: it falsifies the proposition as written. Third, the boundedness mismatch (ν is not 1-bounded) needs a trivial rescaling comment.\n\nThe central argument holds up. This paper is for people working on quantitative polynomial Szemerédi theorems in finite fields, and it will be cited for the weighted norm-control result. Send it to peer review. A good referee will ask for the conjugates and a rewritten Section 1.3; neither request should sink the paper.","headline":"The weighted u^{d+1}-norm control in the main theorem is a real advance, but the rational-function application is overclaimed as written and false for constant φ; both are fixable.","tokens_in":10946,"tokens_out":10048,"would_cite":true,"duration_ms":500705,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11B30","11T23"],"pacs":[],"model":"deepseek-v4-flash","headline":"When a weight's low-degree polynomial-phase sums decay by a power of $p$, weighted polynomial averages in finite fields converge to the unweighted average with error $O(p^{-c})$ — enough to count rational-function progressions of any…","keywords":["weighted multilinear averages","polynomial Szemeredi theorem in finite fields","u^s-norm control","rational function progressions","exponential sums","box norm control","quantitative L2 convergence"],"falsifier":"Take a genuinely fractional rational function, for instance $\\varphi(y)=y/(y^2+1)$ over $\\mathbb{F}_p$, fix a nonzero polynomial $P$ with zero constant term, and compute the maximum over $\\xi\\in\\mathbb{F}_p$ of $p^{-1/2}\\bigl|\\sum_{y\\in\\mathbb{F}_p}e_p(\\xi P(\\varphi(y)))\\bigr|$ for a growing sequence of primes; if this maximum tends to infinity, the uniform exponential-sum premise behind Proposition 1.4 fails and the rational-function corollary needs extra hypotheses. For the main theorem, a directly falsifying configuration would be a strongly $(d+1)$-uniform $\\theta$ and 1-bounded functions for which the $L^2$ discrepancy in Theorem 1.2 does not decay as any power of $p$.","tokens_in":9662,"feed_emoji":"🔢","tokens_out":33548,"duration_ms":318317,"temperature":0.7,"pith_summary":"This paper studies averages of the form $\\mathbb{E}_{y\\in\\mathbb{F}_p}\\theta(y)\\prod_{i=1}^k f_i(x+P_i(y)v_i)$ over the finite field $\\mathbb{F}_p$, where the $P_i$ are linearly independent polynomials of degree at most $d$ and the $v_i$ are nonzero vectors. The main result, Theorem 1.2, states that whenever the weight $\\theta$ is 1-bounded and has power-saving decay of all its polynomial-phase exponential sums of degree at most $d$, this weighted average differs from the unweighted product $\\mathbb{E}_y\\theta(y)\\prod_{i=1}^k\\mathbb{E}_{n_i}f_i(x+n_i v_i)$ by an $L^2$ error whose square is $O_{P,\\theta}(p^{-c})$, uniformly over all 1-bounded functions. Because the error is a power of the field size, the convergence statement becomes quantitative counting machinery: it yields a weighted multidimensional polynomial pattern-counting identity (Theorem 1.3), and — via a classical exponential-sum bound for rational phases — an asymptotic formula for the rational-function progressions $x, x+P_1(\\varphi(y))v_1,\\dots,x+P_k(\\varphi(y))v_k$, whose density consequence is that every subset of $\\mathbb{F}_p^D$ of size at least $C p^{D-c}$ contains such a configuration for any nonzero rational $\\varphi$. A sympathetic reader should care because the uniformity hypothesis is one checkable property of the weight alone, and the rational-function application reaches configurations of arbitrary length, beyond the three-term cases accessible by earlier methods.","feed_headline":"Exponential-sum decay drives weighted averages down to $O(p^{-c})$","feed_subtitle":"One checkable condition on the weight turns convergence into power-saving counts of arbitrarily long configurations.","key_machinery":"The carrying object is the $u^s$-norm, the supremum of $\\bigl|\\mathbb{E}_{y\\in\\mathbb{F}_p}\\theta(y)e_p(-P(y))\\bigr|$ over integer polynomials $P$ of degree at most $s-1$, together with Proposition 3.2, which asserts that for the generalized weighted operators $\\mathcal{G}^\\theta_{l,k}(x)=\\mathbb{E}_y\\theta(y)\\prod_{i=1}^l f_i(x+P_i(y)v_i)\\prod_{i=l+1}^k e_p(P_i(y)\\xi_i)$, one has $\\|\\mathcal{G}^\\theta_{l,k}\\|_{L^2(\\mathbb{F}_p^D)}^{O_P(1)}\\le\\|\\theta\\|_{u^{d+1}(\\mathbb{F}_p)}+O_P(p^{-c_l})$, uniformly in the phases $\\xi_i$. The proof alternates two steps: a box-norm control lemma bounds the $L^2$ norm of the operator by the box norm of an associated dual function $F^\\theta_{l,k}$, and a degree-lowering induction expresses the dual function's directional frequency coefficient along $v_l$ in terms of the $(l-1)$-term operator, so the induction hypothesis applies without ever raising the degree threshold $d+1$. The base case is a direct frequency-orthogonality computation, and at every stage the unweighted multidimensional polynomial pattern-counting theorem is used as a black box. The design choice that makes arbitrary $k$ work is that the control is entirely by the $u^{d+1}$-norm of the weight, not by a uniformity norm of the functions $f_i$.","core_discovery":"The paper's central claim is Theorem 1.2: let $\\theta:\\mathbb{F}_p\\to\\mathbb{C}$ be 1-bounded and strongly $(d+1)$-uniform, meaning $\\|\\theta-\\mathbb{E}\\theta\\|_{u^{d+1}(\\mathbb{F}_p)}\\le C p^{-c}$, where the $u^s$-norm is the supremum of $\\bigl|\\mathbb{E}_y\\theta(y)e_p(-P(y))\\bigr|$ over integer polynomials $P$ of degree at most $s-1$. Then for every 1-bounded $f_1,\\dots,f_k$ and every sufficiently large prime $p$, $\\mathbb{E}_x\\bigl|G^\\theta_k(x)-\\mathbb{E}_y\\theta(y)\\prod_{i=1}^k\\mathbb{E}_{n_i}f_i(x+n_i v_i)\\bigr|^2=O_{P,\\theta}(p^{-c})$, where $G^\\theta_k(x)=\\mathbb{E}_y\\theta(y)\\prod_{i=1}^k f_i(x+P_i(y)v_i)$. The same estimate yields Theorem 1.3, the weighted multidimensional pattern-counting identity, and, as a direct application of a classical exponential-sum bound for rational phases, Proposition 1.4: the rational-function configuration $x,x+P_1(\\varphi(y))v_1,\\dots,x+P_k(\\varphi(y))v_k$ with nonzero $\\varphi\\in\\mathbb{Q}(y)$ has the same asymptotic count as the unweighted independent-variable average, up to $O_{P,\\varphi}(p^{-c})$ and with the average excluding poles. The intended reading is that the $u^{d+1}$-norm of the weight is the single obstruction to quantitative convergence: once its low-degree polynomial-phase exponential sums decay by a power of $p$, everything else in the average is determined.","pith_inferences":["Read as a criterion, the theorem says that exhibiting any concrete weight with provable power-saving $u^{d+1}$-decay — a purely exponential-sum calculation — immediately yields a quantitative weighted pattern-counting statement; the paper proves the implication and demonstrates it for rational-function phases, but does not attempt to verify the criterion for other arithmetic weights such as short-","A natural next step is to verify directly that the rational-phase exponential-sum bound holds uniformly over all auxiliary frequencies, since the proof of Proposition 1.4 (Section 1.3) needs the $O(p^{1/2})$ estimate to hold with constants independent of the frequencies; this verification would either complete the rational-function application as stated or delimit it.","The same two-step mechanism of box-norm control followed by degree-lowering should plausibly transfer to configurations with distinct rational functions $\\varphi_i(y)$ in each slot, or to weights built from normalized multiplicative characters, whenever the corresponding uniform exponential-sum estimates hold; these extensions are not attempted in the paper."],"forward_implications":["Quantitative convergence: for any 1-bounded weight with power-saving decay in the $u^{d+1}$-norm, the weighted average $\\mathbb{E}_y\\theta(y)\\prod_i f_i(x+P_i(y)v_i)$ agrees with $\\mathbb{E}_y\\theta(y)\\prod_i\\mathbb{E}_{n_i}f_i(x+n_i v_i)$ up to a fixed power of $p$ in $L^2$, uniformly over all 1-bounded functions.","Weighted counting identity: Theorem 1.3 asserts the weighted configuration count equals $(\\mathbb{E}_y\\theta(y))$ times the unweighted count up to $O_{P,\\theta}(p^{-c})$; when the weight's average is bounded away from zero, the same density thresholds as the unweighted theorem apply.","Rational-function progressions of any length: Proposition 1.4 gives the asymptotic count for $x, x+P_1(\\varphi(y))v_1,\\dots,x+P_k(\\varphi(y))v_k$ with nonzero rational $\\varphi$, averaging away from poles, extending beyond the three-term cases treated earlier.","Density consequence: any subset of $\\mathbb{F}_p^D$ with size at least $C p^{D-c}$ contains a nontrivial configuration of that rational-function form, with the constants depending only on the data."],"supporting_citations":[{"why":"Supplies the unweighted multidimensional polynomial pattern-counting theorem (Theorem 3.1), the black box that the weighted proof reduces to at every induction step.","marker":"[Kuc24a]"},{"why":"Motivates the central idea of controlling the averages by the $u^s$-norm of the weight rather than by a uniformity norm of the functions themselves.","marker":"[Ter24]"},{"why":"Provides the degree-lowering induction framework (box-norm control plus complexity reduction) that Proposition 3.2 adapts to the weighted multidimensional setting.","marker":"[Pel19]"},{"why":"The quantitative weighted convergence result in the ergodic setting of which Theorem 1.2 is presented as a finite-field version.","marker":"[FK05]"},{"why":"Poses the rational-function progression problem and supplies the exponential-sum bound for rational-function phases used in the proof of Proposition 1.4.","marker":"[BC17]"},{"why":"Provides Lemma 6.1, which converts the asymptotic formula of Proposition 1.4 into the density statement of Corollary 1.5.","marker":"[Kuc24b]"}],"fun_headline_variants":["Weighted multilinear averages: u^s-norm decay yields O(p^{-c}) error","Strong (d+1)-uniformity of weight ensures power-saving pattern counts","One u^s-norm check controls all weighted polynomial averages in finite fields","Rational function progressions: asymptotics from weight's u^{d+1}-norm decay","Power-saving bounds for weighted averages and rational pattern counts"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the classical bound on exponential sums with rational-function phases holds uniformly as the auxiliary frequencies range over all of $\\mathbb{F}_p$, with a fixed power-saving error and constants independent of the frequencies, so that the $p^{-c}$ error of the main theorem survives the summation over all frequencies; the proof of Proposition 1.4 in Section 1.3 invokes this uniformity as a direct application rather than verifying it.","fun_headline_variants_meta":{"raw":{"variants":["Weighted multilinear averages: u^s-norm decay yields O(p^{-c}) error","Strong (d+1)-uniformity of weight ensures power-saving pattern counts","One u^s-norm check controls all weighted polynomial averages in finite fields","Rational function progressions: asymptotics from weight's u^{d+1}-norm decay","Power-saving bounds for weighted averages and rational pattern counts"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00052,"raw_usage":{"total_tokens":2599,"prompt_tokens":1107,"completion_tokens":1492,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":723,"completion_tokens_details":{"reasoning_tokens":1389}},"tokens_in":723,"tokens_out":1492,"duration_ms":14822,"temperature":1.0,"reasoning_tokens":1389,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:58:36.229879+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a genuinely fractional rational function, for instance $\\varphi(y)=y/(y^2+1)$ over $\\mathbb{F}_p$, fix a nonzero polynomial $P$ with zero constant term, and compute the maximum over $\\xi\\in\\mathbb{F}_p$ of $p^{-1/2}\\bigl|\\sum_{y\\in\\mathbb{F}_p}e_p(\\xi P(\\varphi(y)))\\bigr|$ for a growing sequence of primes; if this maximum tends to infinity, the uniform exponential-sum premise behind Proposition 1.4 fails and the rational-function corollary needs extra hypotheses. For the main theorem, a directly falsifying configuration would be a strongly $(d+1)$-uniform $\\theta$ and 1-bounded functions for which the $L^2$ discrepancy in Theorem 1.2 does not decay as any power of $p$.","supporting_citations":[],"review_version":1}