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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})).

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load-bearing objection 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.

arxiv 2607.05124 v1 pith:BHKZ4U5J submitted 2026-07-06 math.CA

A Roth theorem in mathbb R² and a related ergodic theorem

Danqing He , Xinyu Zhu This is my paper
classification math.CA MSC 42B2037A3011B30
keywords quantitative Roth theoremtwo-dimensional polynomial patternsbilinear Sobolev inequalitysublevel set estimatepointwise ergodic averagespolynomial configurations
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

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}] imes[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.

Core claim

Every measurable set E ⊂ [0,N^{2}] imes[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.

What carries the argument

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.

Load-bearing premise

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.

What would settle it

Exhibit a positive-density set in [0,N^{2}] imes[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.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 5 minor

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}] imes[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).

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.

minor comments (5)
  1. Page 1, title and abstract: the arXiv identifier appears as 2607.05124; confirm that this is the intended number before publication.
  2. Section 2.1, display after (8): the factor 2^{2k'} I is written without parentheses; a minor typesetting clarification would improve readability.
  3. 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.
  4. 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.
  5. 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.

Circularity Check

0 steps flagged

No circularity: the Roth and ergodic theorems are derived from a self-contained bilinear estimate whose only novel input is an independent sublevel-set counting argument.

full rationale

The derivation chain is linear and non-circular. Theorem 1 is reduced by a standard density-increment / pigeonhole argument (Section 2.1) to the bilinear Sobolev bound of Theorem 4; Theorem 3 is reduced by Calderón transference and known maximal inequalities to the same bound. Theorem 4 is obtained from the more general Theorem 5 by a smooth cut-off that removes a set of small measure (Section 3). Theorem 5 is reduced, via a frequency decomposition (Lemma 3.1) and TT* expansion, to two estimates: an L1 bound for the flat piece T♭ (Section 4) that uses only Plancherel, support restrictions (Lemma 4.1) and the Fourier-side control already encoded in Lemma 3.1(iii), and an L1 bound for the sharp piece T♯ (Section 5) that is controlled by the purely combinatorial sublevel-set cardinality #I ≲ R^{3}λ^{4γ-δ̃} of Theorem 7. The latter is proved by contradiction via pigeonholing and a Jacobian lower bound |det J(Φ)| ≳ λ^{-8δ̃} that follows from the non-degeneracy conditions built into the support of the multiplier; no parameter is fitted to data, no uniqueness theorem is imported from the authors’ prior work, and no ansatz is smuggled in by citation. All external references (Bourgain, Bergelson–Leibman, Christ–Durcik–Roos, etc.) supply either comparison statements or standard tools that are independent of the target pattern. Consequently the claimed quantitative Roth theorem and the pointwise ergodic theorem stand or fall with the analytic estimates proved in the paper itself.

Axiom & Free-Parameter Ledger

2 free parameters · 3 axioms · 0 invented entities

The paper rests on standard Fourier-analytic machinery and a short list of previously established maximal and pointwise ergodic inequalities; the only ad-hoc ingredients are the concrete phase function and the Jacobian lower bound forced by the pattern.

free parameters (2)
  • exponent hierarchy (γ,δ1,δ2,δ3,δ4,κ,σ,δ̃)
    Small positive numbers chosen once and for all so that all error terms are negative powers of λ; explicit numerical values appear in Remark 3 but are not fitted to data.
  • absolute constants C,K,c appearing in Cl≤2^{Cl} and δ(ε)≳e^{-e^{cε^{-3}}}
    Existential constants produced by the estimates; their precise values are never needed.
axioms (3)
  • standard math Plancherel theorem, non-stationary phase, inverse-function theorem, Bernstein inequalities
    Used throughout Sections 3–5 for Fourier support control and Jacobian estimates.
  • domain assumption Calderón transference principle and the continuous maximal ergodic inequalities of [29]
    Invoked in the proof of the pointwise ergodic theorem (Section 2.2) to reduce to the Euclidean bilinear estimate.
  • ad hoc to paper The Jacobian lower bound |det J(F)|≥A^{-1} on the support of the cut-off r (condition (4))
    Forced by the concrete polynomial pattern; without it the change-of-variables and inverse-function arguments fail.

pith-pipeline@v1.1.0-grok45 · 36169 in / 2287 out tokens · 25599 ms · 2026-07-11T08:26:36.625457+00:00 · methodology

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We prove a quantitative Roth theorem in the plane for the two-dimensional polynomial pattern $(x_1,x_2), (x_1,x_2)+(t_1,t_2), (x_1,x_2)+(t_1^2+t_2^2,t_1^3+t_2^3)$. A pointwise convergence result for the associated polynomial ergodic average is also obtained. A new bilinear Sobolev improving estimate serves as the primary analytic tool, derived from a new sublevel set estimate.

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