An Asymmetric Bound for Sum of Distance Sets

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 $|E|\cdot |F|\gg q^{d+{1}/{3}}$, then $|\Delta (E)+\Delta (F)|>q/2$. They also proved that the threshold $q^{d+{1}/{3}}$ 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.