Distance sets over arbitrary finite fields

Abstract

In this paper, we study the Erdős distinct distances problem for Cartesian product sets in the setting of arbitrary finite fields. More precisely, let ${\mathbb{F}}_q$ be an arbitrary finite field and $A$ be a set in ${\mathbb{F}}_q$. Suppose $|A\cap (aG\left)\right|\le |G{|}^{1/2}$ for any subfield $G$ and $a\in {\mathbb{F}}_q^{\ast }$, then $$\left|{\Delta }_{{\mathbb{F}}_q}\right(A^2\left)\right|=\left|\right(A-A{)}^2+(A-A{)}^2|\gg |A{|}^{1+\frac{1}{21}}.$$ Using the same method, we also obtain some results on sum–product type problems.