On the two-parameter Erdős-Falconer distance problem over finite fields (2101.10959v1)

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.