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 Fdq, 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 Fdq, 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.