Büchi automata are the natural extension of finite automata, also called finite state machines, to a model of computation that accepts infinite length inputs. We say a subset $X$ of the reals is $r$-regular if there is a Büchi automaton that accepts (one of) the base-$r$ representations of every element of $X$, and rejects the base-$r$ representations of each element in its complement. We can analogously define $r$-regular subsets of higher arities, and these sets often exhibit fractal-like behavior--e.g., the Cantor set is 3-regular. There are compelling connections between fractal geometry and Büchi automata, and we will consider them from the perspective of model theory. In this talk, we will give a characterization of when different notions of fractal dimension do or do not agree for definable sets in an expansion of the real ordered additive group by a ternary predicate with a remarkable connection to Büchi automata.

This is joint work with C. Schulz.