A Polynomial Roth Theorem for Corners in Finite Fields

Abstract

We prove a Roth type theorem for polynomial corners in the finite field setting. Let $\ddot{I}{•}_1$ and $\ddot{I}{•}_2$ be two polynomials of distinct degree. For sufficiently large primes $p$, any subset $A\subset {\mathbb{F}}_p\times {\mathbb{F}}_p$ with $|A|>p^{2-\frac{1}{16}}$ contains three points $(x_1,x_2),(x_1+\ddot{I}{•}_1(y),x_2),(x_1,x_2+\ddot{I}{•}_2(y))$. 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.

12 pages. Minor changes for the final version of the paper