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 $\phi_1$ 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.