Fractal dimensions of $k$-automatic sets

TL;DR

This paper characterizes the fractal dimensions of $k$-automatic sets recognized by automata, linking automata theory, fractal geometry, and model theory, and provides algorithms for calculating these dimensions.

Contribution

It introduces a novel characterization of fractal dimensions of $k$-automatic sets using automaton structure and entropy, with algorithmic methods for their computation.

Findings

Provides an algorithmic description of fractal dimensions for $k$-automatic sets.

Connects automata structure with fractal geometric properties.

Lays groundwork for applications in model theory and definability.

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.