Unit Distances in Three Dimensions

We show that the number of unit distances determined by n points in ℝ3 is O(n3/2), slightly improving the bound of Clarkson, Edelsbrunner, Guibas, Sharir and Welzl [5], established in 1990. The new proof uses the recently introduced polynomial partitioning technique of Guth and Katz [12]. While this paper was still in a draft stage, a similar proof of our main result was posted to the arXiv by Joshua Zahl [28

Unit Distances in Three Dimensions

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].Comment: 13 page

The number of unit distances is almost linear for most norms

We prove that there exists a norm in the plane under which no n-point set determines more than O(n log n log log n) unit distances. Actually, most norms have this property, in the sense that their complement is a meager set in the metric space of all norms (with the metric given by the Hausdorff distance of the unit balls)

The density of planar sets avoiding unit distances

By improving upon previous estimates on a problem posed by L. Moser, we prove a conjecture of Erd\H{o}s that the density of any measurable planar set avoiding unit distances cannot exceed $1/4$. Our argument implies the upper bound of $0.2470$.Comment: 24 pages, 6 figures. Final version, to appear in Mathematical Programmin