Erdös distance problem in vector spaces over finite fields

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