A Quadratic Vinogradov Mean Value Theorem in Finite Fields

Abstract

Let $p$ be a prime, let $s\ge 3$ 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 $A\times A$ and a special family of modular hyperbolae.

23 pages; added a reference