On dp-minimal expansions of the integers II

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