Abstract: 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 k-regular if there is a Büchi automaton that accepts (one of) the base-k representations of every element of X, and rejects the base-k representations of each element in its complement. These sets often exhibit fractal-like behavior--e.g., the Cantor set is 3-regular. In this talk, we will contrast the expansion of the real additive group by a predicate for a k-regular set with the same kind of expansion of the real field. We will also discuss how entropy, a measure of complexity coming from information theory, yields a model-theoretic characterization of such expansions. This is joint work with Jason Bell.