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\[D(\mathcal{P}_1,\mathcal{P}_2) = \begin{cases}\Omega(m^{1/2}n^{1/2}\log^{-1/2}n), \ \ & \mbox{ when } m = \Omega(n^{1/2}\log^{-1/3}n), \\\Omega(m^{1/3}n^{1/2}), \ \ & \mbox{ when } m=O(n^{1/2}\log^{-1/3}n). \end{cases}\]This generalizes work of Pohoata and Sheffer, and complements work of Pach and de Zeeuw.