  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.
