A new bound on Erdős distinct distances problem in the plane over prime fields

Abstract

In this paper we obtain a new lower bound on the Erdős distinct distances problem in the plane over prime fields. More precisely, we show that for any set $A\subset {\mathbb{F}}_p^2$ with $|A|\le p^{7/6}$, the number of distinct distances determined by pairs of points in $A$ satisfies $$|\Delta (A)|\gg |A{|}^{\frac{1}{2}+\frac{149}{4214}}.$$ Our result gives a new lower bound of $|\Delta (A)|$ in the range $|A|\le p^{1+\frac{149}{4065}}$. The main tools we employ are the energy of a set on a paraboloid due to Rudnev and Shkredov, a point-line incidence bound given by Stevens and de Zeeuw, and a lower bound on the number of distinct distances between a line and a set in ${\mathbb{F}}_p^2$. The latter is the new feature that allows us to improve the previous bound due Stevens and de Zeeuw.

V2: some typos have been fixed