  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.
