Self-similar Delone sets and Pisot numbers

Abstract

We consider Delone point patterns with self-similarity. Under mild conditions, the similarity factor is a Pisot number if and only if the pattern is uniformly discrete. The classical case is a Meyer set $\Lambda$ with $\Lambda \supset \theta \Lambda$ for some $\theta >1,$ for which $\theta$ must be a Pisot number or a Salem number. When $\Lambda$ contains several similar copies of itself, the case of a Salem number drops out for $\theta <2.$ On the other hand, strictly self-similar patterns with a Pisot factor must be Meyer sets. Various examples are given.