论文标题
$ K $ - 自动套件的分形尺寸
Fractal dimensions of $k$-automatic sets
论文作者
论文摘要
本文旨在基于自动机理论，单词，分形几何学和模型理论之间出现的广泛联系。本文中的结果为“ $ k $自动”集的分形几何形状，$ [0,1]^d $的子集建立了一个表征。构建此特征的主要工具包括常规语言的熵和自动机的挖掘结构。通过分析这种结构的紧密连接组件，我们对盒子计数尺寸，豪斯多夫尺寸和豪斯多夫的算法描述进行了算法描述。在实际添加剂组的模型理论扩展中的确定性应用也是如此。
This paper seeks to build on the extensive connections that have arisen between automata theory, combinatorics on words, fractal geometry, and model theory. Results in this paper establish a characterization for the behavior of the fractal geometry of "$k$-automatic" sets, subsets of $[0,1]^d$ that are recognized by Büchi automata. The primary tools for building this characterization include the entropy of a regular language and the digraph structure of an automaton. Via an analysis of the strongly connected components of such a structure, we give an algorithmic description of the box-counting dimension, Hausdorff dimension, and Hausdorff measure of the corresponding subset of the unit box. Applications to definability in model-theoretic expansions of the real additive group are laid out as well.
