Sphere-and-Point Incidence Relations in High Dimensions with Applications to Unit Distances and Furthest-Neighbor Pairs

Abstracl. For n points in three-dimensional Euclidean space, the number of unit distances is shown to be no more than cn ~/~. Also, we prove that the number of furthest-neighbor pairs for n points in 3-space is no more than cn ~ , provided no three points are collinear. Both these results follow from the following incidence relation of spheres and points in 3-space. Namely, the number of incidences between n points and t spheres is at most en4/~t~'5 if no three points are collinear and n a/'~ > t > n ~/4 The proof is based on a point-and-line incidence relation established by Szemer6di and Trotter. Analogous versions for higher dimensions are also given.