Distinct distances from three points

Abstract

Let $p_1,p_2,p_3$ be three non-collinear points in the plane, and let $P$ be a set of $n$ other points in the plane. We show that the number of distinct distances between $p_1,p_2,p_3$ and the points of $P$ is $\Omega(n^{6/11})$, improving the lower bound $\Omega(n^{0.502})$ of Elekes and Szab\'o \cite{ESz} (and considerably simplifying the analysis).