Let $\mathcal{ R}$ be an expansion of the ordered real additive group. When $\mathcal{ R}$ is o-minimal, it is known that either $\mathcal{ R}$ defines an ordered field isomorphic to $(\mathbb{ R},<,+,\cdot)$ on some open subinterval $I\subseteq \mathbb{ R}$, or $\mathcal{ R}$ is a reduct of an ordered vector space. We say $\mathcal{ R}$ is field-type if it satisfies the former condition. In this paper, we prove a more general result for arbitrary expansions of $(\mathbb{ R},<,+)$. In particular, we show that for expansions that do not define dense $\omega$-orders (we call these type A expansions), an appropriate version of Zilber's principle holds. Among other things we conclude that in a type A expansion that is not field-type, every continuous definable function $[0,1]^m \to \mathbb{ R}^n$ is locally affine outside a nowhere dense set.
Categories: math.LO