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 $$\mathbb {F}_q^d$$ , and $$\Delta (E), \Delta (F)$$ be corresponding distance sets. We prove that if $$|E||F|\ge Cq^{d+\frac{1}{3}}$$ for a sufficiently large constant C, then the set $$\Delta (E)+\Delta (F)$$ 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 $${\mathbb {F}}_q^d,$$ it is shown that the exponent $$d+\frac{1}{3}$$ can be improved to $$d-\frac{1}{6}.$$ 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.