Let $\mathbb{ F}_q$ be a finite field of order $q$ and $\mathcal{ E}$ be a set in $\mathbb{ F}_q^d$. The distance set of $\mathcal{ E}$, denoted by $\Delta(\mathcal{ E})$, is the set of distinct distances determined by the pairs of points in $\mathcal{ E}$. Very recently, Iosevich, Koh, and Parshall (2018) proved that if $|\mathcal{ E}|\gg q^{ d/2}$, then the quotient set of $\Delta(\mathcal{ E})$ satisfies \[\left\vert\frac{ \Delta(\mathcal{ E})}{ \Delta(\mathcal{ E})}\right\vert=\left\vert \left\lbrace\frac{ a}{ b}\colon a, b\in \Delta(\mathcal{ E}), b\ne 0\right\rbrace\right\vert\gg q.\] In this paper, we break the exponent $d/2$ when $\mathcal{ E}$ is a Cartesian product of sets over a prime field. More precisely, let $p$ be a prime and $A\subset \mathbb{ F}_p$. If $\mathcal{ E}=A^d\subset \mathbb{ F}_p^d$ and $|\mathcal{ E}|\gg p^{ \frac{ d}{ 2}-\varepsilon}$ for some $\varepsilon>0$, then we have \[\left\vert\frac{ \Delta(\mathcal{ E})}{ \Delta(\mathcal{ E})}\right\vert, ~\left\vert \Delta(\mathcal{ E})\cdot \Delta(\mathcal{ E})\right\vert \gg p.\] Such improvements are not possible over arbitrary finite fields. These results give us a better understanding about the structure of distance sets and the Erd\H{ o}s-Falconer distance conjecture over finite fields.
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