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Preprint Number 2179

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2179. Alexi Block Gorman and Christian Schulz
Fractal dimensions of k-automatic sets

Submission date: 5 May 2022


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.

Mathematics Subject Classification: 03D05 (Primary) 28A80 (Secondary)

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Full text arXiv 2205.02915: pdf, ps.

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