On the two-parameter Erdős-Falconer distance problem over 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 ${\Delta }_{d,d}(E)=\left\{\left(\Vert x_1-y_1\Vert ,\Vert x_2-y_2\Vert \right):(x_1,x_2),(y_1,y_2)\in E\right\}$. 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.