An asymmetric bound for sum of distance sets

Abstract

For $E\subset {\mathbb{F}}_q^d$, let $\Delta (E)$ denote the distance set determined by pairs of points in $E$. By using additive energies of sets on a paraboloid, Koh, Pham, Shen, and Vinh (2020) proved that if $E,F\subset {\mathbb{F}}_q^d$ are subsets with $|E||F|\gg q^{d+\frac{1}{3}}$ then $|\Delta (E)+\Delta (F)|>q/2$. They also proved that the threshold $q^{d+\frac{1}{3}}$ is sharp when $|E|=|F|$. In this paper, we provide an improvement of this result in the unbalanced case, which is essentially sharp in odd dimensions. The most important tool in our proofs is an optimal $L^2$ restriction theorem for the sphere of zero radius.