Büchi automata are the natural extension of finite automata 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. In this talk, we will examine the interactions between automata theory, model theory, and fractal geometry. We will generalize the notion of sparsity for finite automata to Büchi automata, and demonstrate how this notion gives rise to a dividing line in terms of both fractal geometry and neostability. This is joint work with Jason Bell.