Distribution of distances in positive characteristic

Abstract: Let $\mathbb{F}_q$ be an arbitrary finite field, and $\mathcal{E}$ be a set of points in $\mathbb{F}_q^d$. Let $\Delta(\mathcal{E})$ be the set of distances determined by pairs of points in $\mathcal{E}$. By using the Kloosterman sums, Iosevich and Rudnev proved that if $|\mathcal{E}|\ge 4q^{\frac{d+1}{2}}$, then $\Delta(\mathcal{E})=\mathbb{F}_q$. In general, this result is sharp in odd-dimensional spaces over arbitrary finite fields. In this paper, we use the recent point-plane incidence bound due to Rudnev to prove that if $\mathcal{E}$ has Cartesian product structure in vector spaces over prime fields, then we can break the exponent $(d+1)/2$, and still cover all distances. We also show that the number of pairs of points in $\mathcal{E}$ of any given distance is close to its expected value.