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

Abstract

An analog of the Falconer distance problem in vector spaces over finite fields asks for the threshold α > 0 such that ${|\Delta(E)| \gtrsim q}$ whenever ${|E| \gtrsim q^{\alpha}}$, where ${E \subset {\mathbb {F}}_q^d}$, the d-dimensional vector space over a finite field with q elements (not necessarily prime). Here ${\Delta(E)=\{{(x_1-y_1)}^2+\dots+{(x_d-y_d)}^2: x,y \in E\}}$. Iosevich and Rudnev (Trans Am Math Soc 359(12):6127–6142, 2007) established the threshold ${\frac{d+1}{2}}$, 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 ${\tfrac{4}{3}}$, consistent with the corresponding exponent in Euclidean space obtained by Wolff (Int Math Res Not 10:547–567, 1999). The pinned distance set ${\Delta_y(E)=\{{(x_1-y_1)}^2+\dots+{(x_d-y_d)}^2: x\in E\}}$ for a pin ${y\in E}$ 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 ${\tfrac{d+1}{2}}$, 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 ${\Pi_y(E)=\{x\cdot y: x\in E\}}$. Under the additional assumption that the set E has Cartesian product structure we improve the pinned threshold for both distances and dot products to ${\frac{d^2}{2d-1}}$. The pinned dot product result for Cartesian products implies the following sum-product result. Let ${A\subset \mathbb F_q}$ and ${z\in \mathbb F^*_q}$. If ${|A|\geq q^{\frac{d}{2d-1}}}$ then there exists a subset ${E'\subset A\times \dots \times A=A^{d-1}}$ with ${|E'|\gtrsim |A|^{d-1}}$ such that for any ${(a_1,\dots, a_{d-1}) \in E'}$,

$$ |a_1A+a_2A+\dots +a_{d-1}A+zA| > \frac{q}{2}$$

where ${a_j A=\{a_ja:a \in A\},j=1,\dots,d-1}$. 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 ${|E| \gtrsim q^{\alpha}}$. Here the authors improve on known results (for k > 3) using Fourier analytic methods, showing that α may be taken to be ${\frac{d+k}{2}}$.