Distinct Distances Between a Circle and a Generic Set

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 $\min (\mathrm{\mid }S\mathrm{\mid }\mathrm{\mid }P{\mathrm{\mid }}^{1\mathrm{/}4-\epsilon },\mathrm{\mid }S{\mathrm{\mid }}^{2\mathrm{/}3}\mathrm{\mid }P{\mathrm{\mid }}^{2\mathrm{/}3},\mathrm{\mid }S{\mathrm{\mid }}^2,\mathrm{\mid }P{\mathrm{\mid }}^2)$. This builds on work of Pach and De Zeeuw, Bruner and Sharir, McLaughlin and Omar and Mathialagan on distances between pairs of sets.