Fractal dimensions of $k$-automatic sets

Alexi Block Gorman; Christian Schulz

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arXiv · 2205.02915

Fractal dimensions of $k$-automatic sets

Alexi Block Gorman·Christian Schulz

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Abstract

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 B\"uchi 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.

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BibTeX RIS

Alexi Block Gorman, Christian Schulz. 2022-05-05. Fractal dimensions of $k$-automatic sets. https://arxiv.org/abs/2205.02915

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