We calculate the growth rate of the complexity function for polytopal cut and project sets. This generalizes work of Julien where the almost canonical condition is assumed. The analysis of polytopal cut and project sets has often relied on being able to replace acceptance domains of patterns by so-called cut regions. Our results correct mistakes in the literature where these two notions are incorrectly identified. One may only relate acceptance domains and cut regions when additional conditions on the cut and project set hold. We find a natural condition, called the quasicanonical condition, guaranteeing this property and demonstrate by counterexample that the almost canonical condition is not sufficient for this. We also discuss the relevance of this condition for the current techniques used to study the algebraic topology of polytopal cut and project sets.

中文翻译：

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具有多面体窗口的剪切和投影集 I：复杂性

我们计算了多面体切割和项目集的复杂度函数的增长率。这概括了 Julien 的工作，其中假设了几乎规范的条件。多面体切割和项目集的分析通常依赖于能够用所谓的切割区域替换模式的接受域。我们的结果纠正了文献中错误识别这两个概念的错误。只有当剪辑和项目集的附加条件成立时，才能将接受域和剪辑区域联系起来。我们找到了一个自然条件，称为准规范条件，保证了这一性质，并通过反例证明了几乎规范条件对此是不充分的。