Self-similar Delone sets and Pisot numbers

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 $Λ$ with $Λ\supset θΛ$ for some $θ>1,$ for which $θ$ must be a Pisot number or a Salem number. When $Λ$ contains several similar copies of itself, the case of a Salem number drops out for $θ<2.$ On the other hand, strictly self-similar patterns with a Pisot factor must be Meyer sets. Various examples are given.