Lately exciting new results concerning distinct distances are constantly being discovered. Marcos Charalambides just uploaded this paper to arXiv. The main result of the paper is the following.
Theorem 1 (Charalambides). Consider a curve of degree , and let be a set of points that are contained in . If no irreducible component of is an algebraic helix, then the number of distinct distances that are determined by is (where the constant of proportionality depends on and ).
Algebraic helices are lines and geodesics of the Clifford torus, which is the surface that is parameterized as
where and are constants. One reason why I find this result quite interesting is its connection to characterizing sets that determine a small number of distinct distances (say, distinct distances; see a brief discussion in this post). Only a few properties of such sets are known. Theorem 1 implies that if is a planar -point set that determines distinct distances, then no constant degree curve, except possibly for planar algebraic helices, can contain points of .
Charalambides states that “the proof relies on a link to a certain structural rigidity question on curves”. I still need time to digest how this proof works, but I am always excited to see new connections between combinatorial geometry problems and other parts of math!