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

TL;DR

This paper advances the understanding of distance and dot product sets in finite fields, improving threshold exponents for various geometric configurations and establishing analogs of Euclidean results.

Contribution

It provides new bounds for pinned distance sets, dot product sets, and simplices in finite fields, extending Euclidean geometric measure theory to finite field settings.

Findings

Improved exponent for distance sets in 2D to 4/3.

Finite field analog of Peres-Schlag result for pinned distance and dot product sets.

Enhanced bounds for simplices in finite fields using Fourier analysis.

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 $E \subset {\Bbb F}_q^d$, 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 $\tfrac{4}{3}$, 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 $\tfrac{d+1}{2}$ 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 $\frac{d+k}{2}$.