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⊂F2p with |A|≤p7/6 and p≡3mod4, the number of distinct distances determined by pairs of points in A satisfies Our result gives a new lower bound of |Δ(A)| in the range |A|≤p1+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 F2p. The latter is the new feature that allows us to improve the previous bound due to Stevens and de Zeeuw.