On Superstable Expansions of Free Abelian Groups

We prove that $(\Z,+,0)$ has no proper superstable expansions of finite Lascar rank. Nevertheless, this structure equipped with a predicate defining powers of a given natural number is superstable of Lascar rank $\omega$. Additionally, our methods yield other superstable expansions such as $(\Z,+,0)$ equipped with the set of factorial elements.