On the diameter of separated point sets with many nearly equal distances

A point set is separated if the minimum distance between its elements is 1. We call two real numbers nearly equal if they differ by at most 1. We prove that for any dimension d ≥ 2 and any γ > 0 , if P is a separated set of n points in R d such that at least γ n 2 pairs in ( P 2 ) determine nearly equal distances, then the diameter of P is at least C ( d , γ ) n 2 / ( d − 1 ) for some constant C ( d , γ ) > 0 . In the case of d = 3 , this result confirms a conjecture of Erdős. The order of magnitude of the above bound cannot be improved for any d .