Generalizing a theorem of Bès and Choffrut

Abstract

Bès and Choffrut recently showed that there are no intermediate structures between $(\mathbb{R},<,+)$ and $(\mathbb{R},<,+,\mathbb{Z})$. We prove a generalization: if $\mathcal{R}$ is an o-minimal expansion of $(\mathbb{R},<,+)$ by bounded subsets of Euclidean space then there are no intermediate structures between $\mathcal{R}$ and $(\mathcal{R},\mathbb{Z})$. It follows there are no intermediate structures between $(\mathbb{R},<,+,\sin {|}_{[0,2\pi ]})$ and $(\mathbb{R},<,+,\sin )$.

arXiv comment8 pages. Comments are welcome

References in corpus (3)

• A new dp-minimal expansion of the integers
• Definability and decidability in expansions by generalized Cantor sets
• Theories of real addition with and without a predicate for integers

Cited by in corpus (1)

• Decidability of definability issues in the theory of real addition