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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 Λ with Λ⊃ 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.
Related
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BibTeX
@misc{bandt2026selfsimilar,
title = {{Self-similar Delone sets and Pisot numbers}},
author = {Christoph Bandt and Yves Meyer},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.11867},
eprint = {2608.11867},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.11867},
}
RIS
TY - UNPB
TI - Self-similar Delone sets and Pisot numbers
AU - Christoph Bandt
AU - Yves Meyer
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2608.11867
UR - https://arxiv.org/abs/2608.11867
AB - 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⊃ 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.
ER -