# Multiply tiling Euclidean space by translations of a convex object

# Multiply tiling Euclidean space by translations of a convex object

We study the problem of covering Euclidean space R^d by possibly overlapping translates of a convex body P such that almost every point is covered exactly k times for a fixed integer k. Such a covering of Euclidean space by translations of P is called a k-tiling. Classical tilings by translations (which are 1-tilings in this context) began with the work of the famous crystallographer Fedorov and with the work of Minkowski, who founded the Geometry of Numbers. Some 50 years later Venkov and McMullen gave a complete characterization of all convex objects that 1-tile Euclidean space. Today we know that k-tilings can be tackled by methods from Fourier analysis, though some of their aspects can also be studied using purely combinatorial means. For many of our results there is both a combinatorial proof and a Harmonic analysis proof. For k larger than 1 the collection of convex objects that k-tile is much wider than the collection of objects that 1-tile. So it's a more diverse subject with plenty (infinite families) of examples in R^2

as well. There is currently no complete knowledge of the polytopes that k-tile in dimension 3 or larger, and even in dimension 2 it is still challenging. We will cover both ``ancient'' as well as very recent results concerning 1-tilings and other k-tilings. This is based on some joint work with Nick Gravin, Dmitry Shiryaev, and Mihalis Kolountzakis.