On dp-minimal expansions of the integers

Abstract

We show that if $\mathcal{Z}$ is a dp-minimal expansion of $\left(\mathbb{Z},+,0,1\right)$ that defines an infinite subset of $\mathbb{N}$, then $\mathcal{Z}$ is interdefinable with $\left(\mathbb{Z},+,0,1,<\right)$. As a corollary, we show the same for dp-minimal expansions of $\left(\mathbb{Z},+,0,1\right)$ which do not eliminate ${\exists }^{\infty }$.

arXiv comment21 pages, final version incorporating referee comments