Erdös distance problem in vector spaces over finite fields

Abstract

We study the Erdös/Falconer distance problem in vector spaces over finite fields. Let F q {\mathbb F}_q be a finite field with q q elements and take E ⊂ F q d E \subset {\mathbb F}^d_q , d ≥ 2 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 F q d {\mathbb F}^d_q to provide estimates for minimum cardinality of the distance set Δ ( E ) \Delta (E) in terms of the cardinality of E E . Bounds for Gauss and Kloosterman sums play an important role in the proof.