Unit Distances in Three Dimensions

TL;DR

This paper improves the upper bound on the number of unit distances among n points in three-dimensional space using polynomial partitioning, providing a new proof that slightly advances previous results.

Contribution

It introduces a novel proof employing polynomial partitioning to establish a tighter bound on unit distances in three dimensions.

Findings

Established an O(n^{3/2}) bound for unit distances in R^3

Utilized polynomial partitioning technique of Guth and Katz

Provided a new proof that slightly improves previous bounds

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].