On the quotient set of the distance set

TL;DR

This paper investigates the quotient set of distances in finite field vector spaces, establishing size conditions under which the quotient set covers the entire field or includes all quadratic residues, with results optimal in general.

Contribution

It provides new size thresholds for subsets of finite field vector spaces to ensure their distance quotient sets cover the entire field or include quadratic residues, extending previous understanding.

Findings

For even dimensions, large enough sets produce a quotient set equal to the entire field.

For odd dimensions, the quotient set contains zero and all nonzero quadratic residues.

Results are shown to be optimal in general.

Abstract

Let ${\Bbb F}_q$ be a finite field of order $q.$ We prove that if $d\ge 2$ is even and $E \subset {\Bbb F}_q^d$ with $|E| \ge 9q^{\frac{d}{2}}$ then $$ {\Bbb F}_q=\frac{\Delta(E)}{\Delta(E)}=\left\{ \frac{a}{b}: a \in \Delta(E), b \in \Delta(E) \backslash \{0\} \right\},$$ 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.