Erdös distance problem in vector spaces over finite fields

Iosevich, A.; Rudnev, M.

doi:10.1090/S0002-9947-07-04265-1

Erdös distance problem in vector spaces over finite fields
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by A. Iosevich and M. Rudnev PDF

Trans. Amer. Math. Soc. 359 (2007), 6127-6142 Request permission

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 $E \subset {\mathbb F}^d_q$, $d \ge 2$. We develop a Fourier analytic machinery, analogous to that developed by Mattila in the continuous case, for the study of distance sets in ${\mathbb F}^d_q$ to provide estimates for minimum cardinality of the distance set $\Delta (E)$ in terms of the cardinality of $E$. Bounds for Gauss and Kloosterman sums play an important role in the proof.

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