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, r-regular subsets of the reals, and the directed graph structure of the automata that witness regularity. We will describe how to obtain Hausdorff dimension, box-counting dimension, and Hausdorff measure of an r-regular set in terms of the automaton witnessing r-regularity. We will also briefly discuss applications to model theory. This talk will cover joint work with Christian Schulz.