自相似德洛内集与皮索特数
Self-similar Delone sets and Pisot numbers
AI总结：
本文研究自相似德洛内集，证明温和条件下相似因子为皮索特数等价于模式一致离散，还分析了迈耶集相关情形及给出示例，为相关模式研究提供了理论依据。
AI中文摘要：
我们研究具有自相似性的德洛内点模式。在温和条件下，相似因子为皮索特数当且仅当该模式是一致离散的。经典情形为包含θΛ（θ>1）的迈耶集Λ，此时θ必须为皮索特数或 Salem 数。当Λ包含多个自身的相似副本时，θ<2时 Salem 数情形消失；而具有皮索特因子的严格自相似模式必为迈耶集，文中给出了多个示例。
英文摘要：
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.
