Erdos distance problem in vector spaces over finite fields

Abstract

We study the Erdös/Falconer distance problem in vector spaces over finite fields. Let ${\mathbb{F}}_q$ be a finite field with $q$ elements and take $E\subset {\mathbb{F}}_q^d$, $d\ge 2$. We develop a Fourier analytic machinery, analogous to that developed by Mattila in the continuous case, for the study of distance sets in ${\mathbb{F}}_q^d$ to provide estimates for minimum cardinality of the distance set $\Delta (E)$ in terms of the cardinality of $E$. Kloosterman sums play an important role in the proof.