{"id":"51c4a058-9fad-4532-bfe0-1a84fab4f692","arxiv_id":"2606.30767","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Sparse Roth-type results hold in near-maximal B_h sets; Sidon sets free of equations with five-coefficient zero-sums are small or have bounded-chromatic Cayley graphs, with constructions for the four-coefficient case.","lead":"The paper shows that B_h sets avoiding solutions to linear equations with more than 2h variables must be smaller than the maximum known size by a constant factor, using Fourier pseudorandomness of extremal sets. It also gives conditions under which Sidon sets generate Cayley graphs with bounded or unbounded chromatic number depending on zero-sum subcollections in the forbidden equation.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption flags the pseudorandomness statement, but the paper explicitly claims to prove it rather than assume it. Because the full manuscript is stated to be available and the abstract presents the result as proved, the load-bearing step is internally supported rather than left open. No other technical gap (e.g., in the subdivision-structure case or the chromatic-number constructions) is indicated by the provided material.","tokens_in":1733,"tokens_out":289,"duration_ms":11294,"concrete_test":"Extract the proof that extremal B_h sets are Fourier pseudorandom (the section establishing the key input) and verify that the L^2 or L^infty Fourier bound follows from the B_h sumset condition without additional hypotheses on the ambient group.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim rests on establishing Fourier pseudorandomness for near-maximal B_h sets and then deriving a constant-factor size reduction for equation-free subsets. The abstract states this pseudorandomness is proved as a key input, and the argument is presented as a direct sparse analog of prior Sidon-set results (Conlon-Fox-Sudakov-Zhao, Prendiville). No internal inconsistency, hidden assumption in the linear-equation counting, or unsubstantiated step is visible in the given description.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The paper derives sparse analogs of Roth-type results in near-maximal B_h sets. It shows that a B_h set without pairwise distinct solutions to a linear equation with more than 2h variables must be smaller by a constant factor than the best-known upper bound on |B_h| sets. A key input is the proof that extremal B_h sets are Fourier pseudorandom. For Sidon sets (h=2), it further shows that forbidding a non-translation-invariant equation E with a zero-sum subcollection of at least five coefficients forces either small size or bounded chromatic number in the generated Cayley graph, while constructing large Sidon sets free of equations with four-coefficient zero-sums that generate graphs of unbounded chromatic number. This extends prior Sidon-set work of Conlon-Fox-Sudakov-Zhao and Prendiville, and provides a sparse analog of Liu-Wu-Yang-Zhang on chromatic thresholds.","tokens_in":1806,"tokens_out":451,"duration_ms":14759,"significance":"If the Fourier pseudorandomness of extremal B_h sets holds, the results give a structural dichotomy for equation-free subsets of near-maximal B_h sets and supply new examples of sparse sets with controlled chromatic thresholds. The pseudorandomness input is a reusable tool that strengthens the sparse additive combinatorics toolkit, and the constructions for Sidon sets separate the behavior of different zero-sum lengths in a manner parallel to the dense case.","major_comments":[],"minor_comments":[{"comment":"The abstract and introduction should explicitly state the precise definition of 'pairwise distinct solutions' used in the main theorem (likely in §2 or §3) to avoid ambiguity with the standard B_h definition.","section":null},{"comment":"Notation for the linear equation E and its subdivision structure (mentioned for the asymptotic saving case) should be introduced with a numbered display equation early in the paper.","section":null},{"comment":"The reference list should include the full citation details for Conlon-Fox-Sudakov-Zhao and Prendiville to facilitate comparison with the dense analogs.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive and accurate summary of our manuscript, for highlighting the significance of the Fourier pseudorandomness result as a reusable tool, and for recommending minor revision. The report correctly identifies the extensions of prior work by Conlon-Fox-Sudakov-Zhao, Prendiville, and Liu-Wu-Yang-Zhang.","responses":[],"tokens_in":1373,"tokens_out":90,"duration_ms":13053,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main advance is carrying the Conlon-Fox-Sudakov-Zhao and Prendiville results on Sidon sets over to B_h sets for general h. They show that a near-maximal B_h set without distinct solutions to a linear equation with more than 2h variables must be smaller than the known upper bound by a constant factor. The key step is establishing that those extremal B_h sets are Fourier pseudorandom, which then feeds the counting argument. For the chromatic side they give both a saving when the equation has a certain subdivision structure and explicit constructions of large Sidon sets whose Cayley graphs have unbounded chromatic number while avoiding equations with zero-sum subcollections of four coefficients.\n\nThe pseudorandomness claim is the technical heart and appears to be handled directly rather than assumed. The constructions for the chromatic thresholds look like the right sparse analog of the Liu-Wu-Yang-Zhang work. Citations track the relevant prior results without padding.\n\nThe main soft spot is that the pseudorandomness proof for general h is the load-bearing piece; if it has any gap the size-reduction claim weakens. The abstract is clear on the statements but the full argument needs checking for how the h=2 case lifts. No sign of circularity or invented parameters.\n\nThis is for people already working in additive combinatorics on sparse Roth theorems or on chromatic numbers of Cayley graphs. A reader in that subfield gets concrete extensions and constructions worth knowing. It deserves a serious referee because the generalization is substantive and the methods sit on established ground.","headline":"Extends the Sidon-set Roth analogs to general B_h sets by proving extremal ones are Fourier pseudorandom, plus some new chromatic constructions for non-invariant equations.","tokens_in":2256,"tokens_out":397,"would_cite":false,"duration_ms":13118,"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":"A B_h set without pairwise distinct solutions to a linear equation with more than 2h variables must be smaller by a constant factor than the largest known B_h sets.","keywords":["B_h sets","Sidon sets","linear equations","chromatic number","Fourier pseudorandomness","Roth theorem","additive combinatorics"],"falsifier":"An explicit construction of a B_h set that reaches within a constant factor of the maximum size while avoiding all pairwise distinct solutions to some linear equation with more than 2h variables.","tokens_in":2622,"feed_emoji":"","tokens_out":662,"duration_ms":24971,"temperature":0.7,"pith_summary":"The paper establishes that B_h sets of near-maximum size must contain solutions to linear equations with more than 2h variables, extending Roth-type theorems to this sparse setting. A key step shows that the largest B_h sets are Fourier pseudorandom, which transfers dense results to the sparse case. For Sidon sets, avoiding non-translation-invariant equations with zero-sum subcollections of five or more coefficients forces either a very small set or a Cayley graph of bounded chromatic number. Constructions demonstrate large Sidon sets that avoid equations with four-coefficient zero-sums yet produce graphs of unbounded chromatic number.","feed_headline":"Near-max B_h sets must solve linear equations with over 2h variables","feed_subtitle":"Avoiding pairwise distinct solutions to such equations forces a constant-factor reduction below the known upper bound on size.","key_machinery":"Fourier pseudorandomness of extremal B_h sets, which serves as the input that transfers dense Roth-type results into the sparse B_h setting.","core_discovery":"If a B_h set is free of pairwise distinct solutions to a linear equation with more than 2h variables then it must be a constant factor smaller than the best-known upper bound on the size of any B_h set. Extremal B_h sets are Fourier pseudorandom. If the forbidden equation has a certain subdivision structure, an asymptotic saving is obtained. For Sidon sets, forbidding a non-translation-invariant equation with a zero-sum subcollection of at least five coefficients implies the set is either very small or generates a Cayley graph with bounded chromatic number.","pith_inferences":["The pseudorandomness property may allow similar transfers for other additive-combinatorial statements beyond linear equations.","Chromatic-threshold characterizations could extend from Sidon sets to general B_h sets."],"forward_implications":["Large B_h sets must contain solutions to many linear equations with more than 2h variables.","When the equation has a subdivision structure, the size bound improves to an asymptotic saving.","Sidon sets avoiding equations with zero-sum subcollections of five or more coefficients are either small or generate bounded-chromatic-number Cayley graphs.","Sidon sets avoiding only equations with four-coefficient zero-sums can still be large and generate unbounded-chromatic-number Cayley graphs."],"fun_headline_variants":["B_h sets free of solutions to over 2h-var equations are smaller","Avoiding big linear equations forces smaller B_h sets","Extremal B_h sets are Fourier pseudorandom","Sidon sets with zero-sum equations bound chromatic number"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"Extremal B_h sets are Fourier pseudorandom.","fun_headline_variants_meta":{"raw":{"variants":["B_h sets free of solutions to over 2h-var equations are smaller","Avoiding big linear equations forces smaller B_h sets","Extremal B_h sets are Fourier pseudorandom","Sidon sets with zero-sum equations bound chromatic number"]},"model":"grok-4.3","cost_usd":0.005652,"raw_usage":{"total_tokens":2736,"prompt_tokens":735,"num_sources_used":0,"completion_tokens":59,"cost_in_usd_ticks":56524500,"prompt_tokens_details":{"text_tokens":735,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1942,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":735,"tokens_out":59,"duration_ms":19897,"temperature":1.0,"reasoning_tokens":1942,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-03T22:22:51.007677+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit construction of a B_h set that reaches within a constant factor of the maximum size while avoiding all pairwise distinct solutions to some linear equation with more than 2h variables.","supporting_citations":[],"review_version":2}