论文标题
较高的螺旋delone套件
Higher Dimensional Spiral Delone Sets
论文作者
论文摘要
$ \ mathbb {r}^n $中的DelONE集合是一个集合，即（a）其两个点之间的距离均匀地限制在下面的严格正常常数，并且（b）（b）从集合中的任何点到剩余点的距离都均匀地限制在上面。因此，Delone集是享有良好间距属性的一组点，因此自然而然地出现在准晶体的数学模型中。 在$ \ mathbb {r}^n $中定义一个螺旋式集，为$ \ left \ left \ {\ sqrt [n] {k} \ cdot \ cdot \ boldsymbol {u} _K \ \ right \ right \ right \ right \ right \ right \ ge 1} $， $ \ left（\ boldsymbol {u} _k \ right）_ {k \ ge 1} $是单位球体中的序列$ \ mathbb {s}^{n-1} $。在平面案例$ n = 2 $中，螺旋套件是植物杆菌（研究植物茎上叶子构型的研究）的天然理论模型，此类中的一个重要例子包括向日葵螺旋。 Akiyama，Marklof和Yudin的最新作品提供了对也是DeLone的平面螺旋组的合理特征。在过去几年中，文献中几个地方出现了一个相关的问题，是确定该理论是否可以扩展到更高的维度，尤其是在任何维度中都存在螺旋delone集的存在。 本文通过表征螺旋式集合的Delone属性的填料和覆盖条件来解决这个问题，该属性由球形序列$ \ left（\ boldsymbol {u} _K \ right）_ {k \ ge 1} $。这允许在所有$ n \ ge 2 $中构建$ \ mathbb {r}^n $中的螺旋DeLone集的明确示例，这归结为在$ \ Mathbb {s}^{n-1}中找到一系列积分序列，享受一些最佳分配属性。
A Delone set in $\mathbb{R}^n$ is a set such that (a) the distance between any two of its points is uniformly bounded below by a strictly positive constant and such that (b) the distance from any point to the remaining points in the set is uniformly bounded above. Delone sets are thus sets of points enjoying nice spacing properties, and appear therefore naturally in mathematical models for quasicrystals. Define a spiral set in $\mathbb{R}^n$ as a set of points of the form $\left\{\sqrt[n]{k}\cdot\boldsymbol{u}_k\right\}_{k\ge 1}$, where $\left(\boldsymbol{u}_k\right)_{k\ge 1}$ is a sequence in the unit sphere $\mathbb{S}^{n-1}$. In the planar case $n=2$, spiral sets serve as natural theoretical models in phyllotaxis (the study of configurations of leaves on a plant stem), and an important example in this class includes the sunflower spiral. Recent works by Akiyama, Marklof and Yudin provide a reasonable complete characterisation of planar spiral sets which are also Delone. A related problem that has emerged in several places in the literature over the past fews years is to determine whether this theory can be extended to higher dimensions, and in particular to show the existence of spiral Delone sets in any dimension. This paper addresses this question by characterising the Delone property of a spiral set in terms of packing and covering conditions satisfied by the spherical sequence $\left(\boldsymbol{u}_k\right)_{k\ge 1}$. This allows for the construction of explicit examples of spiral Delone sets in $\mathbb{R}^n$ for all $n\ge 2$, which boils down to finding a sequence of points in $\mathbb{S}^{n-1}$ enjoying some optimal distribution properties.
