Nearly Equal Distances in the Plane

distances determined by them? In particular, what is the maximum number of pairs of points that determine the same distance? Although a lot of progress has been made in this area, we are still very far from having satisfactory answers to the above questions (cf. [EP], [MP], [PA] for recent surveys). Two distances are said to be nearly the same if they differ by at most 1. If all points of a set are close to each other, then all distances determined by them are nearly the same (nearly zero). Therefore, throughout this paper we shall consider only separated point sets P , i.e., we shall assume that the minimal distance between two elements of P is at least 1. In [EMPS] we have shown that the maximum number of times that nearly the same distance can occur among n separated points in the plane is ⌊n2/4⌋, provided that n is sufficiently large. In fact, a straightforward generalization of our argument gives the following.