On Distinct Distances Between a Variety and a Point Set

Abstract

We consider the problem of determining the number of distinct distances between two point sets in ${\mathbb{R}}^2$ where one point set ${\mathcal{P}}_1$ of size $m$ lies on a real algebraic curve of fixed degree $r$, and the other point set ${\mathcal{P}}_2$ of size $n$ is arbitrary. We prove that the number of distinct distances between the point sets, $D({\mathcal{P}}_1,{\mathcal{P}}_2)$, satisfies $D({\mathcal{P}}_1,{\mathcal{P}}_2)=\Omega (m^{1/2}{n}^{1/2}{\log }^{-1/2}n)$ when $m=\Omega (n^{1/2}{\log }^{-1/3}n)$ and $D({\mathcal{P}}_1,{\mathcal{P}}_2)=\Omega (n^{1/2}{m}^{1/3})$ when $m=O(n^{1/2}{\log }^{-1/3}n)$ This generalizes work of Pohoata and Sheffer, and complements work of Pach and de Zeeuw.

8 pages