Abstract
We prove that if a measurable domain tiles R or 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. 1. Introduction. LetE be a measurable set inR n such that 0<jEj<1: We will say that E tiles R n by translations if there is a set T R n such that, up to sets of measure 0, the sets E +t, t2 T , are mutually disjoint and S t2T (E +t) =R n . We call any such T a translation set for E, and write E + T = R n . A tiling E + T = R n is called periodic if it admits a period lattice of rank n; it is a lattice tiling if T itself is a lattice. Here and below, a lattice in R n will always be a set of the form TZ n , where T is a linear transformation of rank n.
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