  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.
