Small sets in Mann pairs

Abstract

Let $\widetilde{\mathcal M}=\langle \mathcal M, G\rangle$ be an expansion of a real closed field $\mathcal M$ by a dense subgroup $G$ of $\langle M^{>0}, \cdot\rangle$ with the Mann property. We prove that the induced structure on $G$ by $\mathcal M$ eliminates imaginaries. As a consequence, every small set $X$ definable in $\mathcal M$ can be definably embedded into some $G^l$, uniformly in parameters. These results are proved in a more general setting, where $\widetilde{\mathcal M}=\langle \mathcal M, P\rangle$ is an expansion of an o-minimal structure $\mathcal M$ by a dense set $P\subseteq M$, satisfying three tameness conditions.