Finite point configurations and projection theorems in vector spaces over finite fields

Abstract

We study a variety of combinatorial distance and dot product related problems in vector spaces over finite fields. First, we focus on the generation of the Special Linear Group whose elements belong to a finite field with q elements. Given A [subset of] Fq, we use Fourier analytic methods to determine how large A needs to be to ensure that a certain product set contains a positive proportion of all the elements of SL₂(Fq). We also study a variety of distance and dot product sets related to the Erd̋os-Falconer distance problem. In general, the Erd̋os-Falconer distance problem asks for the number of distances determined by a set of points. The classical Erdős distance problem asks for the minimal number of distinct distances determined by a finite point set in Rd, where d [is reducible to] 2. The Falconer distance problem, which is the continuous analog of the Erd̋os distance problem, asks to find s₀ [greater than] 0 such that if the Hausdorff dimension of E is greater than s₀, then the Lebesgue measure of [symmetric difference] (E) is positive. A generalization of the Erdős-Falconer distance problem in vector spaces over finite fields is to determine the minimal [alpha] [greater than] 0 such that E contains a congruent copy of every k dimensional simplex whenever [E] [almost equal to] q [alpha]. We improve on known results (for k [greater than] 3) using Fourier analytic methods, showing that [alpha] may be taken to be d+k2 . If E is a subset of a sphere, then we get a stronger result which shows that [alpha] may be taken to be d+k -1 [over] 2.