A Polynomial Roth Theorem for Corners in Finite Fields

TL;DR

This paper establishes a polynomial Roth theorem for corners in finite fields, demonstrating that large subsets contain specific polynomial-configured point triples, extending prior work with advanced algebraic tools.

Contribution

It introduces a new polynomial Roth theorem for corners in finite fields, utilizing deep Weil inequalities and adapting existing arguments to this polynomial setting.

Findings

Large subsets contain polynomial corners with distinct degree polynomials.

The proof relies on Weil type inequalities by N. Katz.

Extends previous results on polynomial configurations in finite fields.

Abstract

We prove a Roth type theorem for polynomial corners in the finite field setting. Let $\phi_1$ and $\phi_2$ be two polynomials of distinct degree. For sufficiently large primes $p$, any subset $ A \subset \mathbb F_p \times \mathbb F_p$ with $ \lvert A\rvert > p ^{2 - \frac1{16}} $ contains three points $ (x_1, x_2) , (x_1 + \phi_1 (y), x_2), (x_1, x_2 + \phi_2 (y))$. The study of these questions on $ \mathbb F_p$ was started by Bourgain and Chang. Our Theorem adapts the argument of Dong, Li and Sawin, in particular relying upon deep Weil type inequalities established by N. Katz.