具有一般多面体窗口的割投影集合的线性重复性刻画
A characterisation of linear repetitivity for cut and project sets with general polytopal windows
AI总结：
本文刻画了割投影集合的线性重复性，将其等价于低复杂度与丢番图条件，并推广至非凸、非齐次及带标记情形，覆盖广义彭罗斯铺砌。
摘要：
割投影方法是Aperiodic Order理论中用于生成具有纯点衍射的准晶体的核心构造。线性重复性（{\bf LR}）是非周期模式理想规律性的一种形式。最近，Koivusalo和本文作者刻画了具有凸多面体窗口的割投影集合的{\bf LR}，其中窗口的支撑超平面与格是共度的，即弱齐次性。对于此类割投影集合，我们证明了{\bf LR}等价于两个性质。一个是低复杂度条件，可通过计算格在内部空间的投影与窗口支撑超平面平行的子空间交集的秩，从割投影数据中确定。第二个条件是格在内部空间的投影是丢番图（或“坏逼近”）的，粗略地说，即全空间中的格点相对于其范数远离物理空间。我们回顾并将这些结果推广到非凸和不相连的多面体窗口，以及具有产生标记点割投影集合的多面体划分的窗口。此外，我们在完全一般情形下获得了{\bf LR}的完整刻画，其中不假设弱齐次性。这里，丢番图性质必须替换为非齐次类比。我们证明了内部空间同构于\\(\mathbb{R}^n \oplus G \oplus \mathbb{Z}^r\\)（其中\\(G\\)为有限阿贝尔群）的割投影方案，在MLD等价意义下，可以约化为内部空间为\\(\mathbb{R}^n\\)的方案，因此我们的结果也涵盖了此类割投影集合，例如（广义）彭罗斯铺砌。
英文摘要：
The cut and project method is a central construction in the theory of Aperiodic Order for generating quasicrystals with pure point diffraction. Linear repetitivity ({\bf LR}) is a form of ideal regularity of aperiodic patterns. Recently, Koivusalo and the present author characterised {\bf LR} for cut and project sets with convex polytopal windows whose supporting hyperplanes are commensurate with the lattice, the weak homogeneity property. For such cut and project sets, we show that {\bf LR} is equivalent to two properties. One is a low complexity condition, which may be determined from the cut and project data by calculating the ranks of the intersections of the projection of the lattice to the internal space with the subspaces parallel to the supporting hyperplanes of the window. The second condition is that the projection of the lattice to the internal space is Diophantine (or `badly approximable'), which loosely speaking means that the lattice points in the total space stay far from the physical space, relative to their norm. We review then extend these results to non-convex and disconnected polytopal windows, as well as windows with polytopal partitions producing cut and project sets of labelled points. Moreover, we obtain a complete characterisation of {\bf LR} in the fully general case, where weak homogeneity is not assumed. Here, the Diophantine property must be replaced with an inhomogeneous analogue. We show that cut and project schemes with internal space isomorphic to \(\mathbb{R}^n \oplus G \oplus \mathbb{Z}^r\), for \(G\) finite Abelian, can, up to MLD equivalence, be reduced to ones with internal space \(\mathbb{R}^n\), so our results also cover cut and project sets of this form, such as the (generalised) Penrose tilings.
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