Simultaneous tiling

Abstract

MIHAIL N. KOLOUNTZAKIS A. We discuss problems of simultaneous tiling. This means that we have an object (set, function) which tiles space with two or more different sets of translations. The most famous problem of this type is the Steinhaus problem which asks for a set simultaneously tiling the plane with all rotates of the integer lattice as translation sets. Dedicated to the memory of Dimitris GatzourasC 1. Tiling by translation. 1 2. Tiling in Fourier space. 2 3. The Steinhaus tiling problem. 3 4. The Steinhaus problem in Fourier space. 4 5. Allowing functions instead of sets in the Steinhaus problem. 4 6. Small diameter of the support: lower bounds. 5 7. A case of large diameter. 6 8. Small volume of the support. 7 9. Small length of the support in d = 1. 9 10. Very small diameter of the support with relations among the lattices. 9 11. Almost matching upper and lower bounds for the diameter, d = 1. 11 References 12 §1. Tiling by translation. For the purposes of this paper 1 tiling is by translation only [16]. We have an object T (the tile) which may be a set or a function on some abelian group G (usually the Euclidean space but it may be Z d or a finite group) which we are F . An L-shaped tile. The red point is the origin translating around by a set of translations Λ, in such a way that everything in the group