Distinct distances from three points

TL;DR

This paper improves the lower bound on the number of distinct distances from three fixed non-collinear points to a set of n points in the plane, advancing understanding in combinatorial geometry.

Contribution

It provides a stronger lower bound of  n^{6/11} for the number of distinct distances, improving previous bounds and simplifying the analysis.

Findings

Established a lower bound of  n^{6/11} on the number of distinct distances.

Improved upon the previous bound of  n^{0.502}.

Simplified the proof technique compared to prior work.

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).