On the structure of distance sets over prime fields

Abstract:

Let $\mathbb {F}_q$ be a finite field of order $q$, and let $\mathcal {E}$ 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 let $A\subset \mathbb {F}_p$. 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 Erdős–Falconer distance conjecture over finite fields.