论文标题
DELONO由方形生成的集合
Delone sets generated by square roots
论文作者
论文摘要
Delone集是局部有限的点集，因此（a）任何两个点都以给定的最小距离隔开，并且（b）有一个给定的半径，因此该半径的每个球都至少包含一个点。重要的例子包括一组Penrose Tilings和其他常规模型集，它们是准晶体的数学模型。在本说明中，我们表明，值$ \ sqrt {n} e^{2πiα\ sqrt {n}} $带有$ n = 1,2,3，\ ldots $是复杂平面中的Delono集合，对于任何$α> 0 $。这补充了Akiyama最近的观察结果（请参见Arxiv：1904.10815），即$ \ sqrt {n} e^{2πiα{n}} $与$ n = 1,2,3，\ ldots $ form n = 1,2,3，\ ldots $ forms a deleone a deleone设置，如果$α$不好，则是由构成理性的。一个关键的区别在于，我们的设置不需要$α$上的二磷酸条件。
Delone sets are locally finite point sets, such that (a) any two points are separated by a given minimum distance, and (b) there is a given radius so that every ball of that radius contains at least one point. Important examples include the vertex set of Penrose tilings and other regular model sets, which serve as a mathematical model for quasicrystals. In this note we show that the point set given by the values $\sqrt{n} e^{2πi α\sqrt{n}}$ with $n=1,2,3,\ldots$ is a Delone set in the complex plane, for any $α>0$. This complements Akiyama's recent observation (see arXiv:1904.10815) that $\sqrt{n} e^{2πi α{n}}$ with $n=1,2,3,\ldots$ forms a Delone set, if and only if $α$ is badly approximated by rationals. A key difference is that our setting does not require Diophantine conditions on $α$.
