{"id":"9e3a6cc0-5c3a-4eb4-9721-2001a80f41fe","arxiv_id":"2607.05124","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.5,"correctness_risk":"low","formal_verification":"none","parameter_count":2,"one_line_summary":"A quantitative Roth theorem holds in R^{2} for the genuinely two-dimensional polynomial pattern (t1,t2) and (t1^{2}+t2^{2}, t1^{3}+t2^{3}), with a matching pointwise ergodic theorem.","lead":"Large sets in the plane must contain the curved three-point pattern (x, x+t, x+(t1^{2}+t2^{2}, t1^{3}+t2^{3})) with quantitative size control. The same analytic estimates yield pointwise convergence of the associated double ergodic averages.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The central claim (Theorem 1) rests on the bilinear estimate of Theorem 5, which is reduced via the standard ♭/♯ decomposition (Lemma 3.1) to an L^{1} bound on the sharp piece T♯. That bound is supplied by the sublevel-set cardinality of Theorem 7. The proof of Theorem 7 proceeds by contradiction: a large set I forces, after two applications of Cauchy–Schwarz and removal of a negligible “degenerate” subset D0, a pair (m2,m'2) with |m2-m'2|≳λ^{γ-δ̃} for which many m1 share the same critical-point coordinates (tm,2,tm',2). The resulting two-dimensional map Φ has Jacobian determinant controlled from below by the non-degeneracy conditions already present in the definition of I and of D', so the inverse-function counting is legitimate. The free exponents in Remark 3 can be chosen small enough that every error term remains negative, and the same counting works uniformly in R≥1. Consequently the analytic core is sound, the reductions are standard, and the reader’s ACCEPT verdict needs no adjustment.","tokens_in":32077,"tokens_out":692,"duration_ms":6224,"concrete_test":"Independently re-derive the lower bound |det J(Φ)|≳λ^{-8δ̃} on S from the explicit form of Ψ(t)=B_R t-P(t) (eq. (50)) without invoking the abstract spectral-radius comparison of Lemma 4.2; if the same power of λ is recovered, the counting in Theorem 7 is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest_assumption correctly flags Theorem 7 (the sublevel-set cardinality #I ≲ R^{3}λ^{4γ-δ̃}) as the most delicate step, but the contradiction argument is internally consistent. After pigeonholing to a large set E'' of m1 with fixed (tm,2,tm',2)=(t0,t'0), the map Φ(s,s')=Ψ(s,t0)-Ψ(s',t'0) has |det J(Φ)|≳λ^{-8δ̃} on the region S defined by the non-degeneracy conditions |s-s'|≳λ^{-2δ̃} and |-3R^{-1}(s+s')+6ss'|≳λ^{-6δ̃} that hold for m1∈E''. Lemma 4.2 then forces the pre-image of each O(λ^{-γ})-ball to lie in O(1) squares of side λ^{-γ+8δ̃}, yielding #E''≲R^{3}λ^{8δ̃}. This contradicts the lower bound #E''≳R^{3}λ^{(1-γ'-σ-4δ̃)/2} once δ̃ is taken smaller than (1-γ'-σ)/24, which is compatible with the free parameters listed in Remark 3. No hidden degeneracy of the phase or Jacobian appears for the concrete pattern P(t)=(t1^{2}+t2^{2},t1^{3}+t2^{3}).","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper establishes a quantitative Roth theorem (Theorem 1) for the genuinely two-dimensional polynomial pattern (x, x+t, x+P(t)) with P(t)=(t_{1}^{2}+t_{2}^{2}, t_{1}^{3}+t_{2}^{3}) in R^{2}: any measurable E ⊂ [0,N^{2}]\times[0,N^{3}] of measure ≥ ε N^{5} contains such a configuration with t_{1},t_{2} > δ(ε)N and δ(ε) ≳ exp(-exp(c ε^{-3})). The same analytic machinery yields pointwise almost-everywhere convergence of the associated continuous double ergodic averages (Theorem 3). The core technical result is a bilinear Sobolev-improving estimate (Theorems 4–5) for the operator T with phase (B_R t, P(t)), obtained via a frequency decomposition into flat/sharp pieces, a TT* argument, non-stationary phase, and a new sublevel-set cardinality bound (Theorem 7).","tokens_in":32504,"tokens_out":806,"duration_ms":6297,"significance":"The work supplies the first quantitative density result for a genuinely two-dimensional polynomial configuration that does not reduce to a one-dimensional pattern (as illustrated in the Appendix). The double-exponential bound, while not optimal, is of the same strength as the best available one-dimensional nonlinear Roth theorems. The accompanying pointwise ergodic theorem advances the continuous analogue of the Bergelson–Leibman conjecture. The new sublevel-set estimate and the careful handling of the non-invertible change of variables for the two-dimensional phase are reusable tools for higher-dimensional polynomial Roth problems. All parameter hierarchies are made fully explicit (Remark 3) and close consistently, which is a notable strength of the presentation.","major_comments":[],"minor_comments":[{"comment":"Page 1, title and abstract: the arXiv identifier appears as 2607.05124; confirm that this is the intended number before publication.","section":null},{"comment":"Section 2.1, display after (8): the factor 2^{2k'} I is written without parentheses; a minor typesetting clarification would improve readability.","section":null},{"comment":"Lemma 4.1 and the subsequent support reduction: the claim that A_m is contained in at most six rectangular boxes is correct, but a one-sentence reminder that the degree-6 eliminant arises from eliminating t_{2} would help the reader.","section":null},{"comment":"Remark 3: the concrete numerical hierarchy (γ=1/100, δ̃=10^{-3}, …) is useful; it would be even clearer if the authors briefly noted that any sufficiently small positive exponents satisfying the listed inequalities work.","section":null},{"comment":"Appendix, Theorem 8: the reduction to the one-dimensional estimate of [16] is clean, but a short sentence explaining why the mixed pattern (t,s^{2}) and (s,t^{2}) is still “essentially one-dimensional” would make the contrast with the main theorem sharper.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is a solid, carefully written contribution that fits well in a top analysis journal. The sublevel-set argument (Theorem 7) is the most delicate step, but the contradiction is internally consistent and the Jacobian lower bound holds for the concrete pattern under consideration. I see no reason to request a major revision."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This is the first quantitative Roth theorem for a genuinely two-dimensional polynomial pattern that does not reduce to one-dimensional cases. They get the configuration x, x+t, x+P(t) with P=(t1^{2}+t2^{2}, t1^{3}+t2^{3}) and a double-exponential density bound, plus the associated pointwise ergodic average. That is real progress inside the continuous nonlinear Roth program.\n\nWhat is new is the bilinear Sobolev estimate (Theorem 5) that handles non-separated support in t, together with the sublevel-set cardinality bound (Theorem 7) that closes the sharp piece. The reduction chain is standard and carefully written: Roth to bilinear, frequency decomposition into flat/sharp/error, TT* plus non-stationary phase, then the counting argument. All free exponents are listed in Remark 3 and close consistently. The appendix shows why several other 2-D patterns collapse to 1-D, so this really is the simplest nontrivial case.\n\nThe soft spot is exactly the one the reader flagged: the contradiction argument for #I in Theorem 7. After pigeonholing to fixed second coordinates, the Jacobian of Φ is bounded below by λ^{-8δ̃} on the non-degenerate region, and Lemma 4.2 forces the pre-images into O(1) small squares. The arithmetic works for the concrete P and the chosen hierarchy, and the stress-test confirms there is no hidden degeneracy. Still, the argument is delicate and the many cut-offs make the constants opaque; a careful referee will want to check the integration-by-parts and the measure of the exceptional sets. That is a technical soft spot, not a structural hole.\n\nCitations are honest and the paper sits cleanly on Durcik–Guo–Roos, Christ–Durcik–Roos, and the earlier 1-D quantitative results. No circularity, no fitted parameters.\n\nThis is for people working on quantitative nonlinear Roth or continuous polynomial ergodic averages. It deserves a serious referee. I would accept it for peer review and would cite the bilinear estimate if I needed a 2-D model case.","headline":"Solid first quantitative Roth for a genuinely 2-D polynomial pattern, with matching pointwise ergodic theorem; the new bilinear estimate and sublevel counting are the real content.","tokens_in":33059,"tokens_out":551,"would_cite":true,"duration_ms":5971,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42B20","37A30","11B30"],"pacs":[],"model":"grok-4.5","headline":"Any dense enough set in a plane rectangle must contain the two-dimensional polynomial pattern (x, x+t, x+(t1^{2}+t2^{2},t1^{3}+t2^{3})).","keywords":["quantitative Roth theorem","two-dimensional polynomial patterns","bilinear Sobolev inequality","sublevel set estimate","pointwise ergodic averages","polynomial configurations"],"falsifier":"Exhibit a positive-density set in [0,N^{2}]\times[0,N^{3}] that avoids the configuration for all t with both components larger than, say, N/log N, or show that the sublevel set I of Theorem 7 can be as large as R^{3} λ^{4}γ for infinitely many scales.","tokens_in":33003,"feed_emoji":"□","tokens_out":792,"duration_ms":6629,"temperature":0.7,"pith_summary":"The paper proves a quantitative Roth-type theorem for a genuinely two-dimensional polynomial configuration in the plane: any measurable set E inside the rectangle [0,N^{2}]\times[0,N^{3}] whose measure is at least a positive fraction ε of the ambient volume must contain three points of the form x, x+t and x+P(t), where P(t)=(t1^{2}+t2^{2},t1^{3}+t2^{3}) and both components of t are larger than a positive multiple of N that depends only on ε (an explicit double-exponential lower bound). The same analytic estimate yields almost-everywhere pointwise convergence of the associated continuous-time double ergodic averages. The result is the first quantitative density theorem for a pattern that cannot be reduced to one-dimensional polynomials, and it is obtained from a new bilinear Sobolev-improving inequality whose proof rests on a carefully counted sublevel-set estimate for the phase of the underlying oscillatory integral.","feed_headline":"Dense plane sets must contain a 2D polynomial triple","feed_subtitle":"First quantitative Roth theorem for a genuinely two-variable polynomial pattern, plus a.e. ergodic convergence","key_machinery":"A bilinear operator T(f1,f2)(x)=∫ f1(x+BRt)f2(x+P(t))r(t)dt, controlled in L^{1} by a Sobolev-improving bound that gains a negative power of the frequency scale of f2; the gain is extracted from a new sublevel-set cardinality estimate (Theorem 7) that counts how often the gradient of the phase can stay small.","core_discovery":"Every measurable set E ⊂ [0,N^{2}]\times[0,N^{3}] with Lebesgue measure at least ε N^{5} contains points x, x+t, x+P(t) with t1,t2 > δ(ε)N, where P(t)=(t1^{2}+t2^{2},t1^{3}+t2^{3}) and δ(ε) ≳ exp(-exp(c ε^{-3})). The same estimate implies that the continuous polynomial ergodic averages AN(f,g) converge almost everywhere for bounded f and g.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Dense plane sets force the 2D polynomial triple x, x+t, x+P(t)","Quantitative Roth in R^{2} for two-variable quadratic-cubic patterns","Thick plane sets contain points x, x+t, x+(t_{1}^{2}+t_{2}^{2}, t_{1}^{3}+t_{2}^{3})","Roth-type result: dense plane measure yields genuine 2D poly configs","Measurable dense plane sets host the specified two-variable progression"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The counting argument that bounds the number of lattice points where the phase gradient is abnormally small must produce a definite power saving; if that counting fails, the L^{1} bound for the high-frequency piece collapses.","fun_headline_variants_meta":{"raw":{"variants":["Dense plane sets force the 2D polynomial triple x, x+t, x+P(t)","Quantitative Roth in R^{2} for two-variable quadratic-cubic patterns","Thick plane sets contain points x, x+t, x+(t_{1}^{2}+t_{2}^{2}, t_{1}^{3}+t_{2}^{3})","Roth-type result: dense plane measure yields genuine 2D poly configs","Measurable dense plane sets host the specified two-variable progression"]},"model":"grok-4.5","effort":"low","cost_usd":0.005862,"raw_usage":{"total_tokens":1471,"prompt_tokens":693,"num_sources_used":0,"completion_tokens":127,"cost_in_usd_ticks":58620000,"prompt_tokens_details":{"text_tokens":693,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":651,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":693,"tokens_out":127,"duration_ms":5545,"temperature":1.0,"reasoning_tokens":651,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-11T08:26:36.625457+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a positive-density set in [0,N^{2}]\times[0,N^{3}] that avoids the configuration for all t with both components larger than, say, N/log N, or show that the sublevel set I of Theorem 7 can be as large as R^{3} λ^{4}γ for infinitely many scales.","supporting_citations":[],"review_version":1}