A tetrachotomy for expansions of the real ordered additive group

TL;DR

This paper classifies expansions of the real ordered additive group into distinct types, showing that non-field-type expansions exhibit tame geometric behavior, with continuous definable functions being locally affine outside small sets.

Contribution

It generalizes the classification of o-minimal expansions to broader classes, establishing a dichotomy and analyzing the geometric properties of definable functions.

Findings

Type A expansions do not define dense ω-orders.

Non-field-type expansions have locally affine continuous functions outside nowhere dense sets.

A generalized Zilber's principle holds for these expansions.

Abstract

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.