On the quotient set of the distance set

Abstract

Let ${\mathbb{F}}_q$ be a finite field of order $q.$ We prove that if $d\ge 2$ is even and $E\subset {\mathbb{F}}_q^d$ with $|E|\ge 9q^{\frac{d}{2}}$ then $${\mathbb{F}}_q=\frac{\Delta (E)}{\Delta (E)}=\left\{\frac{a}{b}:a\in \Delta (E),b\in \Delta (E)\backslash \{0\}\right\},$$ where $$\Delta (E)=\{||x-y||:x,y\in E\},||x||=x_1^2+x_2^2+\cdots +x_d^2.$$ If the dimension $d$ is odd and $E\subset {\mathbb{F}}_q^d$ with $|E|\ge 6q^{\frac{d}{2}},$ then $$\{0\}\cup {\mathbb{F}}_q^{+}\subset \frac{\Delta (E)}{\Delta (E)},$$ where ${\mathbb{F}}_q^{+}$ denotes the set of nonzero quadratic residues in ${\mathbb{F}}_q.$ Both results are, in general, best possible, including the conclusion about the nonzero quadratic residues in odd dimensions.