English

On distinct distances in homogeneous sets in the Euclidean space

Combinatorics 2013-12-17 v4

Abstract

A homogeneous set of nn points in the dd-dimensional Euclidean space determines at least Ω(n2d/(d2+1)/logc(d)n)\Omega(n^{2d/(d^2+1)} / \log^{c(d)} n) distinct distances for a constant c(d)>0c(d)>0. In three-space, we slightly improve our general bound and show that a homogeneous set of nn points determines at least Ω(n.6091)\Omega(n^{.6091}) distinct distances.

Keywords

Cite

@article{arxiv.math/0503443,
  title  = {On distinct distances in homogeneous sets in the Euclidean space},
  author = {J. Solymosi and Cs. D. Toth},
  journal= {arXiv preprint arXiv:math/0503443},
  year   = {2013}
}
R2 v1 2026-07-22T17:17:02.478Z