Abstract
It is shown that every homogeneous set of n points in 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 the above general bound is slightly improved and it is shown that every homogeneous set of n points determines at least \(\Omega(n^{0.6091})\) distinct distances.
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Solymosi, J., Toth, C. Distinct Distances in Homogeneous Sets in Euclidean Space. Discrete Comput Geom 35, 537–549 (2006). https://doi.org/10.1007/s00454-006-1232-4
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DOI: https://doi.org/10.1007/s00454-006-1232-4