A tetrachotomy for expansions of the real ordered additive group

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 $$[0,1]^m \rightarrow \mathbb {R}^n$$ is locally affine outside a nowhere dense set.