We prove that if a measurable domain tiles ℝ or ℝ² 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm160-3-6,
author = {Mihail N. Kolountzakis and Izabella \L aba},
title = {Tiling and spectral properties of near-cubic domains},
journal = {Studia Mathematica},
volume = {162},
year = {2004},
pages = {287-299},
zbl = {1062.52020},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm160-3-6}
}
Mihail N. Kolountzakis; Izabella Łaba. Tiling and spectral properties of near-cubic domains. Studia Mathematica, Tome 162 (2004) pp. 287-299. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm160-3-6/