  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.
