Distinct Distances on a Sphere

We prove that a set of N points on a two dimensional sphere satisfying a discrete energy condition determines at least a constant times N distinct distances. Homogeneous sets in the sense of Solymosi and Vu easily satisfy this condition, as do other sets that in the sense that will be made precise below respect the curvature properties of the sphere. The classical Erdös distance conjecture (EDC) says that a planar point set of cardinality N determines at least a constant times N √ log(N) Euclidean distances. Taking A = [0, √ N ] 2 ∩ Z 2 shows that such an estimate would be best possible ([Erd46]). More precisely, let A ⊂ R 2 with #A = N. Let (0.1) ∆(A) = {x − y : a, b ∈ A}, where z = z 2 1 + z 2 2. See, for example, [PA96] and [PS04] for a description of this beautiful problem and connections with other problems in geometric combinatorics. In spite of many efforts over the past sixty years, the problem remains unsolved. The best known result to date, due to Katz and Tardos ([KT04]), partly based on an ingenuous combinatorial technique by Solymosi and Tóth ([ST01]), gives (0.2) #∆(A) N ≈.86. Here and throughout the paper, a b (a b) means that there exists a universal constant C such that a ≤ Cb (a ≥ Cb) and a ≈ b means that a b and b a. Many of the aforementioned papers on the EDC observe that their method extends to distinct distances on spheres. This is partly due to the fact that many of the principles of planar geometry such as the existence of easily constructed geodesics and bisectors carries over to the spherical setting. In this paper we shall see that the curvature properties of the unit sphere allow for a very efficient accounting procedure for the distance set and lead to a complete resolution of the distance conjecture in this context, under an additional assumption on the set A, that in effect enables one to take advantage of curvature. Our main result is the following.