On dp-minimal expansions of the integers II

Abstract

We first prove that if $\mathcal{Z}$ is a dp-minimal expansion of $\left(\mathbb{Z},+,0,1\right)$ which is not interdefinable with $\left(\mathbb{Z},+,0,1,<\right)$, then every infinite subset of $\mathbb{Z}$ definable in $\mathcal{Z}$ is generic in $\mathbb{Z}$. Using this, we prove that if $\mathcal{Z}$ is a dp-minimal expansion of $\left(\mathbb{Z},+,0,1\right)$ with monster model $G$ such that $G^{00}\neq G^{0}$, then for some $\alpha\in\mathbb{R}\backslash\mathbb{Q}$, the cyclic order on $\mathbb{Z}$ induced by the embedding $n\mapsto n\alpha+\mathbb{Z}$ of $\mathbb{Z}$ in $\mathbb{R}\big/\mathbb{Z}$ is definable in $\mathcal{Z}$. The proof employs the Gleason-Yamabe theorem for abelian groups.