On distinct distances in homogeneous sets in the Euclidean space

J. Solymosi, Cs. D. Toth

Published 2005-03-22, updated 2005-12-08Version 4

A homogeneous set of $n$ points in the $d$-dimensional Euclidean space determines at least $\Omega(n^{2d/(d^2+1)} / \log^{c(d)} n)$ distinct distances for a constant $c(d)>0$. In three-space, we slightly improve our general bound and show that a homogeneous set of $n$ points determines at least $\Omega(n^{.6091})$ distinct distances.