An exciting new result by János Pach and Frank de Zeeuw just appeared on arXiv. The new paper derives both an improvement of the results in the recent paper of Marcos Charalambides (though only for the planar case) and a generalization of the results in a recent paper by Sharir, Solymosi, and myself. Specifically the new paper proves:
Theorem 1. Let be a planar algebraic curve of a constant degree that does not contain any lines or circles. Then any set of points on determines distinct distances.
Theorem 2. Let and be two irreducible planar algebraic curves of constant degrees which are not parallel lines, orthogonal lines, or concentric circles. Then for any points on and points on , the number of distinct distances between the two sets is .
I plan to thoroughly go over the paper this week, and perhaps write a more detailed post about it at some point.