Multiplicative structure in stable expansions of the group of integers

Abstract

We define two families of expansions of $(\mathbb{Z},+)$ by unary predicates, and prove that their theories are superstable of $U$-rank $\omega $. The first family consists of expansions $(\mathbb{Z},+,A)$, where $A$ is an infinite subset of a finitely generated multiplicative submonoid of $\mathbb{Z}^{+}$. Using this result, we also prove stability for the expansion of $(\mathbb{Z},+)$ 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.