A quadratic Vinogradov mean value theorem in finite fields

Abstract

Let p be a prime, let $s \geq 3$ be a natural number and let $A \subseteq \mathbb{F}_p$ be a non-empty set satisfying $|A| \ll p^{1/2}$. Denoting $J_s(A)$ to be the number of solutions to the system of equations $$ \sum_{i=1}^{s} (x_i - x_{i+s}) = \sum_{i=1}^{s} (x_i^2 - x_{i+s}^2) = 0, $$ with $x_1, \dots, x_{2s} \in A$, our main result implies that $$ J_s(A) \ll |A|^{2s - 2 - 1/9}. $$ This can be seen as a finite field analogue of the quadratic Vinogradov mean value theorem. Our techniques involve a variety of combinatorial geometric estimates, including studying incidences between Cartesian products A × A and a special family of modular hyperbolae.