A sharp exponent on sum of distance sets over finite fields

Abstract

We study a variant of the Erdős–Falconer distance problem in the setting of finite fields. More precisely, let E and F be sets in Fqd, and Δ (E) , Δ (F) be corresponding distance sets. We prove that if |E||F|≥Cqd+13 for a sufficiently large constant C, then the set Δ (E) + Δ (F) covers at least a half of all distances. Our result in odd dimensional spaces is sharp up to a constant factor. When E lies on a sphere in Fqd, it is shown that the exponent d+13 can be improved to d-16. Finally, we prove a weak version of the Erdős–Falconer distance conjecture in four-dimensional vector spaces for multiplicative subgroups over prime fields. The novelty in our method is a connection with additive energy bounds of sets on spheres or paraboloids. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature.