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arXiv · 2608.11867

Self-similar Delone sets and Pisot numbers

Christoph Bandt·Yves Meyer
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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.

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Christoph Bandt, Yves Meyer. 2026-08-12. Self-similar Delone sets and Pisot numbers. https://arxiv.org/abs/2608.11867

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