A tetrachotomy for expansions of the real ordered additive group

Let R be an expansion of the ordered real additive group. When R is o-minimal, it is known that either R defines an ordered field isomorphic to (R, < , + , ·) on some open subinterval I⊆ R, or R is a reduct of an ordered vector space. We say R is field-type if it satisfies the former condition. In this paper, we prove a more general result for arbitrary expansions of (R, < , +). In particular, we show that for expansions that do not define dense ω-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→ Rⁿ is locally affine outside a nowhere dense set.