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 $ω$. 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 $(λ_n)_{n=0}^\infty\subseteq\mathbb{R}^+$ such that $\{\frac{λ_n}{λ_m}:m\leq n\}$ is closed and discrete.

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