Erdos distance problem in vector spaces over finite fields

Abstract

We study the Erd\"os/Falconer distance problem in vector spaces over finite fields. Let ${\Bbb F}_q$ be a finite field with $q$ elements and take $E \subset {\Bbb F}^d_q$, $d \ge 2$. We develop a Fourier analytic machinery, analogous to that developed by Mattila in the continuous case, for the study of distance sets in ${\Bbb F}^d_q$ 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.