An Asymmetric Bound for Sum of Distance Sets

Abstract

For E ⊂ Fqd, let Δ(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 ⊂ Fqdare subsets with |E|·|F| >> qd+1/3, then |Δ(E) + Δ(F)|>q/2. They also proved that the threshold qd+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 L2 restriction theorem for the sphere of zero radius.