来自斯图姆晶格的非周期平铺集
Aperiodic tile sets from Sturmian lattices
AI总结：
研究基于二次斜率的斯图姆单词构造非周期平铺集，核心方法是给出显式算法，关键要素有斯图姆晶格和德洛内集的有界位移等价性，能生成基础缩放常数为实二次域单位的非周期平铺集。
AI中文摘要：
我们给出了一种基于二次斜率的斯图姆单词构造非周期平铺集的显式算法。该方法适用于任何二次无理斜率，并且我们可以生成一个非周期平铺集，其基础缩放常数是任何实二次域的单位。我们的构造中有两个关键要素。第一个是“斯图姆晶格”，它是在一个名为史密斯龟的非周期单瓦片出现的由斯图姆单词生成的有趣网格结构。第二个是德洛内集的有界位移等价性，它在这个构造中起着核心作用。完整版本给出了斯图姆晶格的分类和完整证明。
英文摘要：
We give an explicit algorithm to construct aperiodic tile sets based on Sturmian words of quadratic slopes. The method works for any quadratic irrational slope, and we can produce an aperiodic tile set whose underlying scaling constant is a unit of any real quadratic field. There are two key ingredients in our construction. The first one is the ``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. A classification of Sturmian lattices and complete proofs are given in the full version.
