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 $\left|E\right|\ge 9q^{d∕2}$ then

$${\mathbb{F}}_q=\frac{\mathrm{\Delta }\left(E\right)}{\mathrm{\Delta }\left(E\right)}=\left\{\frac{a}{b}:a\in \mathrm{\Delta }\left(E\right),b\in \mathrm{\Delta }\left(E\right)\setminus \left\{0\right\}\right\},$$

where

$$\mathrm{\Delta }\left(E\right)=\left\{\parallel x-y\parallel :x,y\in E\right\},\parallel x\parallel =x_1^2+x_2^2+\cdots +x_d^2.$$

If the dimension $d$ is odd and $E\subset {\mathbb{F}}_q^d$ with $\left|E\right|\ge 6q^{d∕2}$, then

$$\left\{0\right\}\cup {\mathbb{F}}_q^{+}\subset \frac{\mathrm{\Delta }\left(E\right)}{\mathrm{\Delta }\left(E\right)},$$

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.