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 ϕ1 and ϕ2 be two polynomials of distinct degree. For sufficiently large primes p, any subset A ⊂ F p × F p with | A | > p 2 − 1 16 contains three points ( x 1 , x 2 ) , ( x 1 + ϕ 1 ( y ) , x 2 ) , ( x 1 , x 2 + ϕ 2 ( y ) ) . The study of these questions on 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.