Erdös distance problem in vector spaces over finite fields

Abstract

Abstract. We study the Erdös/Falconer distance problem in vector spaces over finite fields. Let F q be a finite field with q elements and takeWe develop a Fourier analytic machinery, analogous to that developed by Mattila in the continuous case, for the study of distance sets in F d q to provide estimates for minimum cardinality of the distance set ∆(E) in terms of the cardinality of E. Bounds for Gauss and Kloosterman sums play an important role in the proof.

Sets with integral distances in finite fields

“…From Lemma 1 and the Weil bound of Kloosterman and Salie sums (see [12, Theorem 11.11 and Lemma 12.4]), we immediately obtain that (see also [11…”

“…Next, we need some results about "Q-spheres" in vector spaces over finite fields which have been used in [11]. Given a non-degenerate quadratic form Q on F n q given by (2), for t ∈ F q we denote by S Q (t) the "Q-sphere"…”

“…There are also several other combinatorial objects to which the results and ideas of [11] can be applied.…”

“…We remark that it follows from [9, Theorem 1.3] (which is a more explicit form of some results of [11]) that the largest independent set of any graph G t is of size at most 4q (n+1)/2 . See also [8], where pseudo-random properties and the diameter of these graphs are studied.…”

“…See, for example, [5,11,19,21] and the references therein for the description of various aspects of this area and recent progress. In this paper we investigate the finite field analog of the well-known problem concerning point sets in R n with pairwise integral Euclidean distances.…”