标题： 斯特姆利亚格子和非周期性瓷砖集合 显示英文标题

标题： Sturmian lattices and Aperiodic tile sets

摘要： 我们基于二次斜率的Sturmian词给出了非周期性瓷砖集合。 该方法适用于任何二次无理斜率，并且我们可以生成无限多个非周期性瓷砖集合，其底层缩放常数是任何实二次域的单位。 我们的构造有两个关键要素。 第一个是“Sturmian格子”，这是一种由Sturmian词生成的有趣的网格结构，在一种非周期性单块瓷砖称为Smith Turtle中出现。 第二个是Delone集的有界位移等价性，这在该构造中起着核心作用。 显示英文摘要

摘要： We give aperiodic tile sets based on Sturmian words of quadratic slopes. The method works for any quadratic irrational slope and we can produce infinitely many aperiodic tile sets whose underlying scaling constant is a unit of any real quadratic field. There are two key ingredients in our construction. The first one is ``Sturmian lattices'', an interesting grid structure generated by Sturmian words that emerged in an aperiodic monotile called Smith Turtle. The second is the bounded displacement equivalence of Delone sets, which plays a central role in this construction.