{"id":"0f41eccc-1beb-459a-9a55-c71ff06385ae","arxiv_id":"2604.27501","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Any subset A of a finite field F of odd characteristic with size at least C times |F| to the 5/6 must contain a nontrivial quadratic progression x, x+y, x+y^2.","lead":"The paper proves that subsets of finite fields of odd characteristic larger than C times the field size to the 5/6 power must contain a quadratic progression of the form x, x+y, x+y squared with y nonzero. A smart generalist might read it to track how simpler exponential sum tools can improve density thresholds in additive combinatorics over finite fields.","discovery_kind":"extension","skeptic_critique":null,"referee_report":{"model":"grok-4.3","summary":"The paper proves that any subset A of a finite field F of odd characteristic with |A| ≥ C |F|^{5/6} contains a nontrivial quadratic progression (x, x+y, x+y²) with y ≠ 0. For prime fields this improves the prior exponent 7/8 due to Kavrut and Wu. The argument relies exclusively on one-variable Weil-type estimates rather than Katz's multivariate exponential-sum bounds. The paper also constructs quadratic-progression-free sets of size c |F|^{2/3} over certain non-prime finite fields.","tokens_in":1622,"tokens_out":650,"duration_ms":105227,"significance":"If correct, the result improves the quantitative threshold for a non-linear Roth theorem in finite fields while simplifying the analytic tools required. The restriction to one-variable Weil bounds makes the proof more elementary and potentially easier to generalize. The matching lower-bound construction in selected non-prime fields shows that 5/6 cannot be improved uniformly across all odd-characteristic fields and clarifies the distinction between prime and composite cases. The work therefore refines the landscape of additive-combinatorial results over finite fields.","major_comments":[{"comment":"§4, the exponential-sum estimate (around Eq. (4.5)): the reduction from the quadratic-progression count to a one-variable character sum must be verified explicitly; the paper should confirm that the relevant polynomial remains of degree 2 after the change of variables and that the Weil bound applies directly without invoking any auxiliary multivariate estimates.","section":"§4"},{"comment":"§5, the progression-free construction: the size c |F|^{2/3} is stated for certain non-prime fields; the precise algebraic condition on F (e.g., existence of a subfield of index 3) and the verification that the constructed set indeed avoids all solutions to x + y² = x + y with y ≠ 0 should be written out in full detail.","section":"§5"}],"minor_comments":[{"comment":"Abstract: the sentence crediting ChatGPT 5.5 for a key idea is atypical in a mathematical abstract and would be more appropriately placed in the acknowledgments.","section":"Abstract"},{"comment":"Introduction: a brief one-sentence recall of the precise statement of the Kavrut–Wu 7/8 result would help readers compare the new exponent directly.","section":"Introduction"},{"comment":"Notation section: the term 'nontrivial' quadratic progression is defined by y ≠ 0, but it should be stated explicitly whether any further degeneracy (e.g., y = 0 or x in a subfield) is excluded.","section":"Notation"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript appears to be a self-contained improvement that avoids heavy machinery; the only potential concern is whether the one-variable Weil application in the density-increment step is written with sufficient explicitness for a referee to check the constant C without re-deriving the estimates."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading, positive assessment, and recommendation of minor revision. The comments identify places where greater explicitness will improve the manuscript. We address each major comment below.","responses":[{"response":"We agree that the reduction step merits an explicit verification. Although the manuscript indicates that only one-variable Weil estimates are used, the change-of-variables computation and the confirmation that the resulting polynomial has degree exactly 2 (with nonzero leading coefficient in odd characteristic) are only sketched. In the revised version we will insert, immediately after Equation (4.5), a self-contained paragraph that carries out the substitution, verifies the degree, and invokes the standard one-variable Weil bound directly. No multivariate estimates are required or used.","revision_made":"yes","referee_comment":"[§4] §4, the exponential-sum estimate (around Eq. (4.5)): the reduction from the quadratic-progression count to a one-variable character sum must be verified explicitly; the paper should confirm that the relevant polynomial remains of degree 2 after the change of variables and that the Weil bound applies directly without invoking any auxiliary multivariate estimates."},{"response":"We accept that the construction section is insufficiently detailed. The sets of size c |F|^{2/3} are built when F admits a subfield of index 3. In the revision we will state this condition explicitly (F is a cubic extension of a subfield K with |F| = |K|^3 and char F odd) and give the precise definition of the set A. We will then supply a complete, self-contained argument showing that A contains no x and y ≠ 0 such that x, x + y, x + y² all lie in A, using the norm or trace properties of the cubic extension to reach a contradiction.","revision_made":"yes","referee_comment":"[§5] §5, the progression-free construction: the size c |F|^{2/3} is stated for certain non-prime fields; the precise algebraic condition on F (e.g., existence of a subfield of index 3) and the verification that the constructed set indeed avoids all solutions to x + y² = x + y with y ≠ 0 should be written out in full detail."}],"tokens_in":1340,"tokens_out":497,"duration_ms":48525,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"Lewko improves the density threshold for quadratic progressions in odd-characteristic finite fields from 7/8 to 5/6 over prime fields. Any set A with |A| at least C times |F| to the 5/6 must contain a nontrivial (x, x+y, x+y^2) with y nonzero. He also constructs progression-free sets of size roughly c |F|^{2/3} in certain non-prime fields. The argument uses only one-variable Weil-type estimates rather than Katz's multivariate bounds from earlier papers.","headline":"Lewko improves the quadratic progression exponent to 5/6 over prime fields using one-variable Weil estimates and adds a 2/3-sized construction in some non-prime fields.","tokens_in":2126,"tokens_out":194,"would_cite":false,"duration_ms":65924,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Any set of size at least C |F|^{5/6} in a finite field of odd characteristic contains a nontrivial quadratic progression x, x+y, x+y².","keywords":["finite fields","quadratic progressions","Roth theorem","exponential sums","Weil estimates","additive combinatorics","density increment","nonlinear configurations"],"falsifier":"An explicit subset A of a large prime field F with |A| > C |F|^{5/6} containing no nontrivial (x, x+y, x+y²) with y ≠ 0 would disprove the main claim.","tokens_in":2523,"feed_emoji":"🧮","tokens_out":775,"duration_ms":83197,"temperature":0.7,"pith_summary":"The paper proves that subsets of finite fields of odd characteristic whose size exceeds a constant times the field size to the 5/6 power must contain three terms in quadratic progression of the form x, x+y, x+y² with y nonzero. This lowers the threshold from the previous 7/8 exponent that held for prime fields. The argument applies a density increment or Fourier analysis that reduces the problem to bounding certain exponential sums, which are controlled using only one-variable Weil-type estimates. The paper also exhibits quadratic-progression-free sets of size c |F|^{2/3} in certain non-prime fields.","feed_headline":"Sets larger than |F|^{5/6} contain quadratic progressions","feed_subtitle":"Improved nonlinear Roth bound for odd-characteristic fields uses one-variable Weil estimates and beats prior 7/8 exponent","key_machinery":"Density increment argument that reduces quadratic-progression detection to exponential sums controlled by one-variable Weil-type estimates.","core_discovery":"Any subset A of a finite field F of odd characteristic with |A| ≥ C |F|^{5/6} contains a nontrivial quadratic progression (x, x+y, x+y²) for some y ≠ 0. For prime fields this improves the previous best exponent of 7/8. The proof proceeds via density increment or Fourier analysis that reduces the detection of the progression to bounding associated exponential sums, which are estimated using only one-variable Weil-type bounds. Over certain non-prime finite fields the paper constructs quadratic-progression-free sets of size c |F|^{2/3}.","pith_inferences":["The gap between the |F|^{2/3} construction and the 5/6 theorem indicates that the exponent can likely be improved further with more refined analysis.","Analogous statements for higher-degree polynomial progressions or in higher-dimensional finite vector spaces may follow from similar exponential-sum control.","The result connects to questions about the maximal size of sets avoiding polynomial configurations, with possible implications for pseudorandomness and coding theory over finite fields."],"forward_implications":["The quantitative threshold guaranteeing a quadratic progression improves from 7/8 to 5/6 when the field is prime.","Only one-variable Weil estimates are required, avoiding the need for Katz's multivariate exponential-sum bounds.","In some non-prime fields the largest sets without quadratic progressions have size at least c |F|^{2/3}, so the true threshold lies between 2/3 and 5/6.","The same density-increment-plus-Weil-estimates approach may apply to other nonlinear configurations in finite fields."],"fun_headline_variants":["Nonlinear Roth theorem improved to 5/6 density in finite fields","Quadratic progressions in sets of size |F|^{5/6} over odd char F","Improved nonlinear Roth for quadratic progressions in finite fields","|F|^{5/6} sets contain quadratic progressions in odd char fields"],"cache_read_input_tokens":64,"weakest_assumption_plain":"One-variable Weil-type estimates suffice to bound the exponential sums that arise when density increment or Fourier analysis is used to detect the quadratic progression, and the field has odd characteristic.","fun_headline_variants_meta":{"raw":{"variants":["Nonlinear Roth theorem improved to 5/6 density in finite fields","Quadratic progressions in sets of size |F|^{5/6} over odd char F","Improved nonlinear Roth for quadratic progressions in finite fields","|F|^{5/6} sets contain quadratic progressions in odd char fields"]},"model":"grok-4.3","cost_usd":0.010367,"raw_usage":{"total_tokens":4483,"prompt_tokens":619,"num_sources_used":0,"completion_tokens":73,"cost_in_usd_ticks":103665500,"prompt_tokens_details":{"text_tokens":619,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3791,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":619,"tokens_out":73,"duration_ms":51976,"temperature":1.0,"reasoning_tokens":3791,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-07T08:26:17.500249+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit subset A of a large prime field F with |A| > C |F|^{5/6} containing no nontrivial (x, x+y, x+y²) with y ≠ 0 would disprove the main claim.","supporting_citations":[],"review_version":1}