Tiling and spectral properties of near-cubic domains
Volume 160 / 2004
Studia Mathematica 160 (2004), 287-299
MSC: 52C20, 42A99.
DOI: 10.4064/sm160-3-6
Abstract
We prove that if a measurable domain tiles ${\mathbb R}$ or ${\mathbb R}^2$ by translations, and if it is “close enough” to a line segment or a square respectively, then it admits a lattice tiling. We also prove a similar result for spectral sets in dimension 1, and give an example showing that there is no analogue of the tiling result in dimensions 3 and higher.
