非周期系统中暴胀与投影集：窗口在平均和衍射中的作用
Inflation versus projection sets in aperiodic systems: The role of the window in averaging and diffraction
摘要：
基于切割和投影方法的Tilings是描述非周期性固体的关键模型系统。通常，晶体学中感兴趣的量涉及在大块上求平均值，并且仅在无限体积极限中被很好地定义。特别是，这是自相关和衍射措施的情况。对于剪切和投影系统，平均可以方便地转移到内部空间，这意味着处理相应的窗口。我们说明了这一点的例子，平均脱壳数的斐波那契瓷砖和审查标准的方法来衍射这个例子。此外，我们讨论了最近的发展，通货膨胀对称的削减和项目结构，这是基于内部对应的重正化上循环。最后，我们简要地回顾了超均匀性的概念，这是最近流行的，它的应用程序的非周期性结构。
英文摘要：
Tilings based on the cut and project method are key model systems for the description of aperiodic solids. Typically, quantities of interest in crystallography involve averaging over large patches, and are well defined only in the infinite-volume limit. In particular, this is the case for autocorrelation and diffraction measures. For cut and project systems, the averaging can conveniently be transferred to internal space, which means dealing with the corresponding windows. We illustrate this by the example of averaged shelling numbers for the Fibonacci tiling and review the standard approach to the diffraction for this example. Further, we discuss recent developments for inflation-symmetric cut and project structures, which are based on an internal counterpart of the renormalisation cocycle. Finally, we briefly review the notion of hyperuniformity, which has recently gained popularity, and its application to aperiodic structures.
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