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 $\alpha >0$ 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 $\Delta (E)=\{{(x_1-y_1)}^2+...+{(x_d-y_d)}^2:x,y\in E\}$. In two dimensions we improve the known exponent to $\frac{4}{3}$, consistent with the corresponding exponent in Euclidean space obtained by Wolff. The pinned distance set ${\Delta }_y(E)=\{{(x_1-y_1)}^2+...+{(x_d-y_d)}^2:x\in E\}$ for a pin $y\in E$ has been studied in the Euclidean setting. Peres and Schlag showed that if the Hausdorff dimension of a set $E$ is greater than $\frac{d+1}{2}$ then the Lebesgue measure of ${\Delta }_y(E)$ 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 $\frac{d^2}{2d-1}$. A generalization of the Falconer distance problem is determine the minimal $\alpha >0$ such that $E$ contains a congruent copy of every $k$ dimensional simplex whenever $|E|\gtrsim q^{\alpha }$. Here the authors improve on known results (for $k>3$) using Fourier analytic methods, showing that $\alpha$ may be taken to be $\frac{d+k}{2}$.

A few corrections

References in corpus (1)

• On The Solvability of Bilinear Equations in Finite Fields