Substitution schemes provide a classical method for constructing tilings of Euclidean space. Allowing multiple scales in the scheme, we introduce a rich family of sequences of tile partitions generated by the substitution rule, which include the sequence of partitions of the unit interval considered by Kakutani as a special case. Using our recent path counting results for directed weighted graphs, we show that such sequences of partitions are uniformly distributed, thus extending Kakutani's original result. Furthermore, we describe certain limiting frequencies associated with sequences of partitions, which relate to the distribution of tiles of a given type and the volume they occupy.

中文翻译：

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由替代方案生成的角谷分区的均匀分布

替代方案为构造欧几里得空间的平铺提供了一种经典方法。允许方案中有多个尺度，我们引入了由替换规则生成的丰富的瓦片分区序列系列，其中包括 Kakutani 认为的单位间隔的分区序列作为特例。使用我们最近对有向加权图的路径计数结果，我们表明这样的分区序列是均匀分布的，从而扩展了 Kakutani 的原始结果。此外，我们描述了与分区序列相关的某些限制频率，这些频率与给定类型的瓦片的分布及其占据的体积有关。