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

On the nonexistence of k-reptile simplices in R3 and R4

A d-dimensional simplex S is called a k-reptile (or a k-reptile simplex) if it can be tiled without overlaps by k simplices with disjoint interiors that are all mutually congruent and similar to S. For d=2, triangular k-reptiles exist for many values of k and they have been completely characterized by Snover, Waiveris, and Williams. On the other hand, the only k-reptile simplices that are known for d≥3, have k=m d, where m is a positive integer. We substantially simplify the proof by Matoušek and the second author that for d=3, k-reptile tetrahedra can exist only for k=m 3. We also prove a weaker analogue of this result for d=4 by showing that four-dimensional k-reptile simplices can exist only for k=m 2.Czech Science FoundationCentre Interfacultaire BernoulliSwiss National Science Foundatio