There are compelling and long-established connections between automata theory and model theory, particularly regarding expansions of Presburger arithmetic by sets whose base-k representations are recognized by an automaton. We call such sets "k-automatic". In this talk, we will characterize the expansions of (N,<,+) by a unary k-automatic set. We can characterize such expansions both in terms of model-theoretic properties, and via a notion of "entropy" coming from information theory. This is joint work with Jason Bell and Chris Schulz.