Multiplicative structure in stable expansions of the group of integers

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 $q\in {\mathbb{N}}_{\ge 2}$. The second family consists of sets $A\subseteq \mathbb{N}$ which grow asymptotically close to a $\mathbb{Q}$-linearly independent increasing sequence $({\lambda }_n{)}_{n=0}^{\mathrm{\infty }}\subseteq {\mathbb{R}}^{+}$ such that $\{\frac{{\lambda }_n}{{\lambda }_m}:m\le n\}$ is closed and discrete.