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On distinct distances in homogeneous sets in the Euclidean space

J. Solymosi·Cs. D. Toth
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Abstract

A homogeneous set of $n$ points in the $d$-dimensional Euclidean space determines at least $Ω(n^{2d/(d^2+1)} / \log^{c(d)} n)$ 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 $Ω(n^{.6091})$ distinct distances.

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J. Solymosi, Cs. D. Toth. 2005-12-08. On distinct distances in homogeneous sets in the Euclidean space. https://doi.org/10.1007/s00454-006-1232-4

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