On the two-parameter Erd\H{o}s-Falconer distance problem over finite fields

TL;DR

This paper investigates the two-parameter Erdős-Falconer distance problem over finite fields, providing new bounds and improvements on the size of sets needed to determine a large portion of possible distances.

Contribution

The paper extends and improves previous bounds on the size of sets required to ensure large two-parameter distance sets over finite fields.

Findings

Improved bounds for the size of E to guarantee large distance sets.

Extended results to higher dimensions and different parameters.

Enhanced understanding of the Erdős-Falconer distance problem in finite fields.

Abstract

Given $E \subseteq \mathbb{F}_q^d \times \mathbb{F}_q^d$, with the finite field $\mathbb{F}_q$ of order $q$ and the integer $d \ge 2$, we define the two-parameter distance set as $\Delta_{d, d}(E)=\left\{\left(\|x_1-y_1\|, \|x_2-y_2\|\right) : (x_1,x_2), (y_1,y_2) \in E \right\}$. Birklbauer and Iosevich (2017) proved that if $|E| \gg q^{\frac{3d+1}{2}}$, then $ |\Delta_{d, d}(E)| = q^2$. For the case of $d=2$, they showed that if $|E| \gg q^{\frac{10}{3}}$, then $ |\Delta_{2, 2}(E)| \gg q^2$. In this paper, we present extensions and improvements of these results.