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arxiv: 2604.27501 · v1 · submitted 2026-04-30 · 🧮 math.NT · math.CO

An improved non-linear Roth-type theorem in finite fields

Mark Lewko This is my paper

Pith reviewed 2026-05-07 08:26 UTC · model grok-4.3

classification 🧮 math.NT math.CO
keywords finite fieldsquadratic progressionsRoth theoremexponential sumsWeil estimatesadditive combinatoricsdensity incrementnonlinear configurations
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The pith

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.

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

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.

Core claim

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}.

What carries the argument

Density increment argument that reduces quadratic-progression detection to exponential sums controlled by one-variable Weil-type estimates.

If this is right

  • 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.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • 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.

Load-bearing premise

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.

What would settle it

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.

read the original abstract

Let $F$ be a finite field of odd characteristic. We prove that any set $A\subset F$ with $|A|\geq C|F|^{5/6}$ contains a nontrivial quadratic progression $(x, x+y, x+y^2), y\neq 0.$ For prime fields, this improves the previous best-known exponent of $7/8$, due to Kavrut and Wu. Unlike some of the previous papers, which rely on Katz's deep multivariate exponential-sum estimates, our argument uses only one-variable Weil-type estimates. We also construct, over certain non-prime finite fields, progression-free sets of size $c|F|^{2/3}$. A key idea in the proof was suggested to the author by ChatGPT 5.5.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

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.

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 (2)
  1. [§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.
  2. [§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.
minor comments (3)
  1. [Abstract] 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.
  2. [Introduction] 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.
  3. [Notation] 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.

Simulated Author's Rebuttal

2 responses · 0 unresolved

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.

read point-by-point responses

  1. Referee: [§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.

    Authors: 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: yes

  2. Referee: [§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.

    Authors: 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: yes

Circularity Check

0 steps flagged

No circularity; proof relies on external standard Weil bounds

full rationale

The derivation proceeds via a density-increment or Fourier-analytic argument that invokes one-variable Weil-type estimates on exponential sums; these bounds are standard results from algebraic geometry and are not derived, fitted, or redefined inside the paper. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the central argument. The claimed exponent improvement follows directly from applying these external bounds to the relevant sums, and the separate construction of progression-free sets over non-prime fields is independent. The paper is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof relies on standard properties of finite fields of odd characteristic and classical Weil bounds; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Finite fields of odd characteristic admit the one-variable Weil estimates needed to control the exponential sums in the argument.
    Invoked to replace the multivariate estimates used in earlier work.

pith-pipeline@v0.9.0 · 5413 in / 1312 out tokens · 93460 ms · 2026-05-07T08:26:17.500249+00:00 · methodology

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Reference graph

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