On Distinct Distances Between a Variety and a Point Set - McLaughlin, Bryce; Omar, Mohamed (The Electronic Journal of Combinatorics, 2022)
We consider the problem of determining the number of distinct distances between two point sets in R^2 where one point set P_1 of size m lies on a real algebraic curve of fixed degree r, and the other point set P_2 of size n is arbitrary. We prove that the number of distinct distances between the point sets, D(P_1,P_2), satisfies \[ D(P_1,P_2) = cases Ω(m^1/2n^1/2^-1/2n), & when m = Ω(n^1/2^-1/3n), Ω(m^1/3n^1/2), & when m=O(n^1/2^-1/3n). cases \] This generalizes work of Pohoata and Sheffer, and complements work of Pach and de Zeeuw.