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

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.

• The paper demonstrates that any finite field subset of size at least C|F|^(5/6) must contain a nontrivial quadratic progression (x, x+y, x+y^2), improving the previous exponent of 7/8.
• It develops a bilinear operator framework alongside Fourier analysis to achieve a sharp decay rate of |F|^(-1/4) without resorting to complex multivariate exponential sum machinery.
• The study presents explicit constructions avoiding these progressions at approximately |F|^(2/3), indicating a natural barrier for current methods in finite field configurations.

Summary of "An improved non-linear Roth-type theorem in finite fields"

Main Contributions

This paper establishes a refined upper bound on the minimal size required for a subset $A \subset F$ of a finite field $F$ (of odd characteristic) to guarantee the presence of a nontrivial quadratic progression $(x, x+y, x+y^2)$, $y \neq 0$, inside $A$. Specifically, it is proved that any $A\subset F$ with $|A|\geq C|F|^{5/6}$ contains such a progression, improving the previous best exponent of $7/8$. The author also provides explicit constructions showing that sets of size about $|F|^{2/3}$ can avoid such progressions in certain non-prime fields. Notably, the proof technique circumvents the use of higher-dimensional exponential sum machinery (notably Katz's results), employing instead only one-variable Weil-type bounds.

Background and Previous Results

The problem is a polynomial variant of Roth's theorem in additive combinatorics, adapted to finite fields and quadratic progressions. Earlier works, starting with Bourgain and Chang (2016), progressively decreased the required density threshold:

• Bourgain and Chang: $|A| \geq c|F|^{14/15}$
• Dong, Li, and Sawin: $F$0
• Kavrut and Wu: $F$1

Those arguments crucially leveraged deep bounds on multidimensional exponential sums via the work of Katz and Deligne. In contrast, this paper achieves improved parameters using only classical one-dimensional Weil bounds, leading to substantially simpler arguments and broader applicability.

Bilinear Operator Framework

Central to the argument is a bilinear averaging operator:

$F$2

The analysis seeks to control the $F$3 norm of $F$4, showing it is sharply bounded by $F$5. This sharp $F$6 decay rate is critical to pushing the progression density threshold down to $F$7.

The argument uses detailed Fourier analysis, reducing the problem to estimating certain sums where the quadratic phase allows the application of Weil's bound. The operator’s structure lets the author avoid Katz's intricate multidimensional estimates, instead iterating bounds in a manner only requiring single-variable exponential-sum control.

Lower Bounds and Extremal Constructions

The paper rigorously analyzes lower bounds and presents explicit constructions of large sets in certain finite fields that avoid quadratic progressions of the form $F$8. These constructions include:

• In quadratic extensions, $F$9, explicit $(x, x+y, x+y^2)$0-lines of size $(x, x+y, x+y^2)$1 that are progression-free.
• In cubic extensions, a counting argument yields $(x, x+y, x+y^2)$2-planes in $(x, x+y, x+y^2)$3 of size $(x, x+y, x+y^2)$4 that exclude such progressions.

These constructions demonstrate that the $(x, x+y, x+y^2)$5 density threshold is a natural barrier, at least without fundamentally new ideas.

Technical Innovations

• Operator Analysis and One-variable Sums: The proof leverages the relationship between the bilinear operator and progression counts, applying Parseval’s and Fourier inversion, and reduces the estimation of key terms to the control of one-variable mixed exponential sums.
• Multiplicative Convolution Structure: The author identifies that the kernel of certain sums can be dealt with via multiplicative convolution, exploiting the group structure and Parseval-type identities in multiplicative characters.
• Use of Weil-type Bounds: By reducing the use of exponential sums to the one-dimensional case, only the classical Weil bound is needed—yielding both technical simplicity and generality.

Implications and Future Directions

The improved density result shifts the threshold for polynomial Roth-type theorems in finite fields closer to the combinatorial obstructions given by explicit constructions. The avoidance of deep multivariate techniques in favor of classical one-variable tools may allow these arguments to generalize to other nonlinear configurations or settings where higher-dimensional estimates are unwieldy or unknown.

The explicit constructions at density $(x, x+y, x+y^2)$6 suggest this may represent a fundamental limit for current methods. Pushing below this exponent, or obtaining matching lower bounds in all finite fields (not just non-prime fields), remains an open challenge.

Theoretical implications include:

• Potential applications to related polynomial Szemerédi-type and higher-degree progression problems.
• Insight into the complexity-theoretic and algebraic barriers for polynomial configuration-free sets over finite fields.

Practically, such results may inform randomness extraction, pseudorandomness constructions, and theoretical aspects of coding theory, where controlling arithmetic progressions and their polynomial analogues is vital.

Conclusion

This work sharpens the quantitative bounds for nonlinear Roth-type theorems in finite fields, establishing a $(x, x+y, x+y^2)$7 exponent threshold and showing that deep multivariate exponential sum machinery is not always necessary. The approach streamlines the proof strategy and nearly matches the known lower bounds evident from explicit constructions. The results provide a blueprint for using classical tools to address modern problems in additive combinatorics and polynomial configuration avoidance.