An asymmetric bound for sum of distance sets

TL;DR

This paper improves bounds on the sum of distance sets in finite fields, especially in unbalanced cases, using advanced restriction theorems, and is nearly sharp in odd dimensions.

Contribution

It provides an improved, essentially sharp bound for the sum of distance sets in unbalanced cases over finite fields, extending prior work.

Findings

Enhanced bounds for sum of distance sets in finite fields

Optimal restriction theorem for sphere of zero radius

Results are nearly sharp in odd dimensions

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.