Unit Distances in Three Dimensions
Combinatorics
2011-07-07 v1
Abstract
We show that the number of unit distances determined by n points in R^3 is O(n^{3/2}), slightly improving the bound of Clarkson et al. established in 1990. The new proof uses the recently introduced polynomial partitioning technique of Guth and Katz [arXiv:1011.4105]. While this paper was still in a draft stage, a similar proof of our main result was posted to the arXiv by Joshua Zahl [arXiv:1104.4987].
Keywords
Cite
@article{arxiv.1107.1077,
title = {Unit Distances in Three Dimensions},
author = {Haim Kaplan and Jiri Matousek and Zuzana Safernova and Micha Sharir},
journal= {arXiv preprint arXiv:1107.1077},
year = {2011}
}
Comments
13 pages