Distinct Distances in Homogeneous Sets in Euclidean Space

It is shown that every homogeneous set of n points in d-dimensional Euclidean space determines at least \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Omega(n^{2d/(d^2+1)} / \log^{c(d)} n)$\end{document} distinct distances for a constant c(d) > 0. In three-space the above general bound is slightly improved and it is shown that every homogeneous set of n points determines at least \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Omega(n^{0.6091})$\end{document} distinct distances.