Sphere-and-point incidence relations in high dimensions with applications to unit distances and furthest-neighbor pairs

Abstract

Forn points in three-dimensional Euclidean space, the number of unit distances is shown to be no more thancn 8/5. Also, we prove that the number of furthest-neighbor pairs forn points in 3-space is no more thancn 8/5, 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 betweenn points andt spheres is at mostcn 4/5 t 4/5 if no three points are collinear andn 3/2>t>n 1/4. The proof is based on a point-and-line incidence relation established by Szemerédi and Trotter. Analogous versions for higher dimensions are also given.