Abstract 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 with is a Delone set in the complex plane, for any . This complements Akiyama’s recent observation (see Spiral Delone sets and three distance theorem (2020), Nonlinearilty, 33(5): 2533–2540) that with 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.

中文翻译：

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由平方根生成的 Delone 集

摘要 Delone 集是局部有限点集，使得 (a) 任意两点相隔一个给定的最小距离，并且 (b) 有一个给定的半径，使得该半径的每个球至少包含一个点。重要的例子包括 Penrose tiles 的顶点集和其他规则模型集，它们用作准晶体的数学模型。在本笔记中，我们展示了由 值给出的点集是复平面中的 Delone 集，对于任何 。这补充了 Akiyama 最近的观察（参见 Spiral Delone 集和三距离定理 (2020), Nonlinearilty, 33(5): 2533–2540），当且仅当 α 被有理数严重逼近时，它形成了 Delone 集。一个关键的区别是我们的设置不需要 α 的丢番图条件。