Multiplicative structure in stable expansions of the group of integers

Abstract: We define two families of expansions of $(\mathbb{Z},+,0)$ by unary predicates, and prove that their theories are superstable of $U$-rank $\omega$. The first family consists of expansions $(\mathbb{Z},+,0,A)$, where $A$ is an infinite subset of a finitely generated multiplicative submonoid of $\mathbb{N}$. Using this result, we also prove stability for the expansion of $(\mathbb{Z},+,0)$ by all unary predicates of the form ${q^n:n\in\mathbb{N}}$ for some $q\in\mathbb{N}{\geq 2}$. The second family consists of sets $A\subseteq\mathbb{N}$ which grow asymptotically close to a $\mathbb{Q}$-linearly independent increasing sequence $(\lambda_n){n=0}^\infty\subseteq\mathbb{R}^+$ such that ${\frac{\lambda_n}{\lambda_m}:m\leq n}$ is closed and discrete.