Pinned distance sets, k-simplices, Wolff’s exponent in finite fields and sum-product estimates

An analog of the Falconer distance problem in vector spaces over finite fields asks for the threshold α > 0 such that whenever , where , the d -dimensional vector space over a finite field with q elements (not necessarily prime). Here . Iosevich and Rudnev (Trans Am Math Soc 359(12):6127–6142, 2007 ) established the threshold , and in Hart et al. (Trans Am Math Soc 363:3255–3275, 2011 ) proved that this exponent is sharp in odd dimensions. In two dimensions we improve the exponent to , consistent with the corresponding exponent in Euclidean space obtained by Wolff (Int Math Res Not 10:547–567, 1999 ). The pinned distance set for a pin has been studied in the Euclidean setting. Peres and Schlag (Duke Math J 102:193–251, 2000 ) showed that if the Hausdorff dimension of a set E is greater than , then the Lebesgue measure of Δ y ( E ) is positive for almost every pin y . In this paper, we obtain the analogous result in the finite field setting. In addition, the same result is shown to be true for the pinned dot product set . Under the additional assumption that the set E has Cartesian product structure we improve the pinned threshold for both distances and dot products to . The pinned dot product result for Cartesian products implies the following sum-product result. Let and . If then there exists a subset with such that for any , where . A generalization of the Falconer distance problem is to determine the minimal α > 0 such that E contains a congruent copy of a positive proportion of k -simplices whenever . Here the authors improve on known results (for k > 3) using Fourier analytic methods, showing that α may be taken to be .