# Small sets in Mann pairs

## TL;DR

This paper proves that in certain o-minimal expansions of real closed fields with dense subgroups, small definable sets can be embedded into finite powers of the subgroup, extending previous results to a more general setting.

## Contribution

It establishes elimination of imaginaries for the induced structure on dense subgroups with the Mann property in a broad o-minimal context, generalizing prior work.

## Key findings

- Small sets in the structure can be embedded into finite powers of the dense subgroup.
- Elimination of imaginaries holds for the induced structure on the dense subgroup.
- Results apply to a general class of o-minimal expansions with tameness conditions.

## 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.