A new bound for the Erdős distinct distances problem in the plane over prime fields
Volume 193 / 2020
Abstract
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 $|A|\le p^{7/6}$ and $p\equiv 3\mod 4$, the number of distinct distances determined by pairs of points in $A$ satisfies $$ |\Delta (A)| \gtrsim |A|^{\frac {1}{2}+\frac {149}{4214}}.$$ Our result gives a new lower bound of $|\Delta {(A)}|$ in the range $|A|\le p^{1+\frac {149}{4065}}$.
The main tools in our method 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 to Stevens and de Zeeuw.
