{"id":"9809a34e-c3d3-4d1c-8a65-ee134b8e1583","arxiv_id":"2605.16216","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Extends Furstenberg-Sárközy to general intersective polynomials h via uniform arithmetic level-d inequality, yielding the best known quasipolynomial density bound.","lead":"This paper adapts the arithmetic level-d inequality to general intersective polynomials and proves a quasipolynomial upper bound on the largest subset of {1..X} without nonzero h(n) differences. A smart generalist might read it to see how density-increment arguments are extended uniformly when the avoided polynomial changes during iteration.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Uniformity of arithmetic level-d inequality may depend on auxiliary polynomial parameters in iteration","rationale":"The reader's weakest assumption directly identifies the step the paper itself presents as its main technical contribution. Because the square case fixes the polynomial while the general case does not, this uniformity is the precise point at which the argument is least secure; verifying the constant dependence settles whether the extension succeeds.","tokens_in":1683,"tokens_out":343,"duration_ms":43475,"concrete_test":"Locate the statement and proof of the uniform arithmetic level-d inequality; extract the explicit dependence of all implied constants on the polynomial coefficients and degree. Then run the density-increment iteration symbolically for h(x)=x^2 and for h(x)=x(x+1), counting the number of steps until the density increment exceeds the initial density; check whether the accumulated loss remains quasipolynomial in log log X.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires a density-increment iteration in which the intersective polynomial changes at each step. For the quasipolynomial bound to survive the full iteration, the arithmetic level-d inequality must hold with constants whose dependence on the current polynomial (degree, leading coefficient, height) remains controlled uniformly across all polynomials generated. If the proof only establishes effectiveness for a fixed bounded class of polynomials and the iteration produces auxiliaries whose parameters grow, the effective density-increment threshold could deteriorate faster than quasipolynomially, collapsing the argument. The abstract flags this uniformity as the key new ingredient, yet the load-bearing step is whether the constants are tracked explicitly enough to guarantee the iteration closes with the stated bound.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper adapts Green and Sawhney's arithmetic level-d inequality to general intersective polynomials h ∈ ℤ[x]. It proves a quasipolynomial upper bound on the size of the largest subset of {1,…,X} whose difference set avoids nonzero values of h(n), via a density-increment iteration in which the underlying polynomial changes at each step. A central contribution is a uniformity statement ensuring the level-d inequality remains effective for all auxiliary polynomials generated during the iteration. The authors also establish smoothly weighted versions of Rice's exponential-sum estimates.","tokens_in":1798,"tokens_out":413,"duration_ms":25862,"significance":"If the uniformity claim holds with explicit parameter dependence, the result supplies the strongest known quantitative bound for sets without intersective polynomial differences and demonstrates that the Green-Sawhney method extends beyond squares. The weighted exponential-sum estimates are likely to be reusable in other arithmetic-progression or polynomial-difference problems.","major_comments":[{"comment":"§3 (Uniformity of the arithmetic level-d inequality): the argument must track the dependence of the implied constants on the degree, leading coefficient, and height of the current auxiliary polynomial explicitly. Without such tracking, it is unclear whether the density-increment threshold remains strong enough for the full iteration to close with a quasipolynomial bound rather than a tower-type loss.","section":"§3"}],"minor_comments":[{"comment":"The statement of the main theorem should include the precise form of the quasipolynomial (e.g., X / (log log X)^c) rather than the generic phrase 'quasipolynomial upper bound'.","section":"Theorem 1.1"},{"comment":"Notation for the weighted exponential sums in §5 should be aligned with the unweighted versions introduced earlier to avoid reader confusion.","section":"§5"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading and constructive comments on our manuscript. We address the major comment below and will incorporate revisions to strengthen the presentation of the uniformity argument.","responses":[{"response":"We thank the referee for this observation. The uniformity statement in Section 3 is formulated so that the arithmetic level-d inequality holds for each auxiliary polynomial arising in the iteration, with the implied constants depending on its degree, leading coefficient, and height. To make this fully explicit and confirm that the density-increment thresholds yield only quasipolynomial losses overall, we will revise the manuscript to record the precise functional dependence of all constants on these parameters throughout the iteration analysis. This will verify that no tower-type losses are introduced, consistent with the claimed quasipolynomial bound.","revision_made":"yes","referee_comment":"[§3] §3 (Uniformity of the arithmetic level-d inequality): the argument must track the dependence of the implied constants on the degree, leading coefficient, and height of the current auxiliary polynomial explicitly. Without such tracking, it is unclear whether the density-increment threshold remains strong enough for the full iteration to close with a quasipolynomial bound rather than a tower-type loss."}],"tokens_in":1275,"tokens_out":271,"duration_ms":44593,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper takes the recent Green-Sawhney quasipolynomial bound for square differences and carries it over to arbitrary intersective h in Z[x]. They obtain the same type of bound for the largest subset of [1,X] whose differences avoid nonzero values of h(n). The central new piece is proving that the level-d inequality remains effective with constants that do not deteriorate when the auxiliary polynomial is replaced at each step of the density increment. They also supply smoothly weighted exponential-sum estimates building on Rice. These two ingredients let the iteration close with a quasipolynomial upper bound, which is currently the strongest quantitative result for this family of problems. The uniformity claim is presented as the main technical contribution, and the abstract makes clear that the argument must track dependence on degree, leading coefficient, and height across the generated auxiliaries. If those dependencies are controlled explicitly enough, the bound survives; otherwise the increment threshold could slip. The weighted estimates appear to be a clean but secondary addition. Readers already comfortable with the square case will see immediately what has been added and where the new bookkeeping sits. The work is aimed at people doing quantitative density-increment arguments in additive combinatorics or ergodic theory. It is worth sending to a serious referee because the uniformity step, if correct, is reusable beyond this specific theorem.","headline":"This extends Green-Sawhney to general intersective polynomials by showing the arithmetic level-d inequality stays uniform when auxiliaries change during iteration.","tokens_in":2296,"tokens_out":336,"would_cite":false,"duration_ms":33271,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[],"headline":"Arithmetic level-d inequality and density increments for intersective polynomials unrelated to RS forcing","alignment":"orthogonal","rationale":"Paper develops quasipolynomial bounds via arithmetic level-d inequality (Theorem A), Rice exponential sums, and iterated density increments on auxiliary polynomials h_ℓ for intersective h. Central objects are Fourier mass on major arcs, smoothly weighted Gauss sums, and uniformity across evolving polynomial coefficients. RS framework (reality_from_one_distinction, Jcost uniqueness via Aczél, AlexanderDuality for D=3, ArithmeticFromLogic recovery of Nat) derives physics constants and 3D spacetime from bare distinction + reciprocal cost; no shared structures, cost functions, periodicity, or parameter-free derivations appear. Domain is additive combinatorics; RS has no theorems on Furstenberg–Sárközy-type avoidance.","tokens_in":70779,"confidence":"high","tokens_out":188,"duration_ms":12095,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"For any intersective polynomial h the largest subset of {1,...,X} without nonzero h(n) differences has a quasipolynomial upper bound on size.","keywords":["Furstenberg-Sarkozy theorem","intersective polynomials","arithmetic level-d inequality","density increment","exponential sums","quasipolynomial bounds","additive combinatorics"],"falsifier":"An explicit construction of a subset A of {1,...,X} with |A| larger than the claimed quasipolynomial bound that still avoids all nonzero h(n) differences, or an auxiliary polynomial arising in the iteration for which the level-d inequality fails to produce a positive density increment.","tokens_in":2601,"feed_emoji":"","tokens_out":657,"duration_ms":53991,"temperature":0.7,"pith_summary":"The paper adapts the arithmetic level-d inequality of Green and Sawhney to prove a quasipolynomial upper bound for the maximal size of a subset of {1,2,...,X} whose differences avoid all nonzero values of a general intersective polynomial h in Z[x]. The central technical step is to run a density-increment iteration in which the underlying polynomial is allowed to change at each stage while still keeping the inequality effective. This produces the strongest quantitative version presently known for the Furstenberg-Sárközy theorem in the polynomial setting. A sympathetic reader cares because the result quantifies how rapidly sets must thin out to avoid structured differences of polynomial type.","feed_headline":"Quasipolynomial bound for sets avoiding polynomial differences","feed_subtitle":"For any intersective h the largest subset of 1 to X without nonzero h(n) differences is bounded by a slow-growing quasipolynomial factor.","key_machinery":"The arithmetic level-d inequality, which supplies a uniform density-increment bound even when the polynomial is updated at each step of the iteration.","core_discovery":"By showing that the arithmetic level-d inequality remains effective uniformly across the sequence of auxiliary polynomials generated by the density-increment iteration, the authors obtain a quasipolynomial upper bound on the size of the largest subset of {1,2,...,X} whose difference set contains no nonzero element of the form h(n) for an arbitrary intersective polynomial h.","pith_inferences":["The uniformity argument may extend to finite families of intersective polynomials or to multidimensional configurations.","Numerical verification for small X and low-degree polynomials could test how close the bound comes to being sharp.","The same uniformity technique might combine with other increment methods to handle still broader classes of configurations."],"forward_implications":["The quasipolynomial bound now holds for every fixed intersective polynomial rather than only for squares.","The iteration proceeds with a changing polynomial without losing the quasipolynomial gain at each step.","Smoothly weighted versions of Rice's exponential-sum estimates are available to support the argument."],"fun_headline_variants":["Quasipolynomial bound for sets avoiding intersective polynomial differences","Arithmetic level-d inequality uniform in density iterations","Bound on sets without h(n) differences is quasipolynomial","Extensions of Furstenberg-Sarkozy via level-d inequality"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The arithmetic level-d inequality remains effective uniformly across all auxiliary polynomials arising in the iteration.","fun_headline_variants_meta":{"raw":{"variants":["Quasipolynomial bound for sets avoiding intersective polynomial differences","Arithmetic level-d inequality uniform in density iterations","Bound on sets without h(n) differences is quasipolynomial","Extensions of Furstenberg-Sarkozy via level-d inequality"]},"model":"grok-4.3","cost_usd":0.010075,"raw_usage":{"total_tokens":4377,"prompt_tokens":640,"num_sources_used":0,"completion_tokens":67,"cost_in_usd_ticks":100753000,"prompt_tokens_details":{"text_tokens":640,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3670,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":640,"tokens_out":67,"duration_ms":43167,"temperature":1.0,"reasoning_tokens":3670,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-19T18:27:12.380056+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit construction of a subset A of {1,...,X} with |A| larger than the claimed quasipolynomial bound that still avoids all nonzero h(n) differences, or an auxiliary polynomial arising in the iteration for which the level-d inequality fails to produce a positive density increment.","supporting_citations":[],"review_version":1}