发表机构
University of Manchester; Ben-Gurion University of the Negev（曼彻斯特大学; 内盖夫本-古里安大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文综述Delone集的双Lipschitz与有界位移等价性，重点研究非周期序相关实例，涉及经典结果、相关判据及替换铺砌、截集等类别的扩展情况。
AI 中文摘要
本文综述了Delone集的双Lipschitz（BL）与有界位移（BD）等价性，重点关注非周期序中出现的实例。在回顾Burago-Kleiner、McMullen和Laczkovich的经典结果后，我们讨论了可求长性、均匀扩散性及BD等价性的判据，以及它们与偏差和点计数估计的关联。随后我们聚焦于与替换 tilings（铺砌）相关的Delone集，这类问题常可通过基础替换规则的组合与谱性质进行研究。我们还考虑了选定类别的截集并简要探讨了超出BL与BD等价性的扩展情况。
英文摘要
We survey biLipschitz (BL) and bounded displacement (BD) equivalence of Delone sets, with an emphasis on examples arising in aperiodic order. After recalling the classical results of Burago-Kleiner, McMullen, and Laczkovich, we discuss criteria for rectifiability, uniform spreadness, and BD equivalence, and how these relate to discrepancy and point-counting estimates. We then focus on Delone sets associated with substitution tilings, where these questions can often be studied through the combinatorial and spectral properties of the underlying substitution rules. We also consider selected classes of cut-and-project sets and briefly discuss extensions beyond BL and BD equivalence.
