Distinct Distances Between a Circle and a Generic Set

Abstract

Let $S$ be a set of points in ${\mathbb{R}}^2$ contained in a circle and $P$ an unrestricted point set in ${\mathbb{R}}^2$. We prove the number of distinct distances between points in $S$ and points in $P$ is at least $\min (|S||P{|}^{1/4-\epsilon },|S{|}^{2/3}|P{|}^{2/3},|S{|}^2,|P{|}^2)$. This builds on work of Pach and De Zeeuw, Bruner and Sharir, McLaughlin and Omar and Mathialagan on distances between pairs of sets.

10 pages