Small sets in Mann pairs

Abstract

Let $\tilde{\mathcal{M}}=⟨\mathcal{M},G⟩$ be an expansion of a real closed field $\mathcal{M}$ by a dense subgroup $G$ of $⟨M^{>0},\cdot ⟩$ 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 $\tilde{\mathcal{M}}=⟨\mathcal{M},P⟩$ is an expansion of an o-minimal structure $\mathcal{M}$ by a dense set $P\subseteq M$, satisfying three tameness conditions.