We consider expansions (M, P) of an o-minimal structure by a dense set P, such that the geometric behavior on the class of all definable sets remains tame. For example, P could be an elementary substructure of M, or an independent set, or a multiplicative subgroup with the Mann property. We prove that every P-small set X can be definably embedded into some cartesian power of P, uniformly in parameters. Together with a structure theorem with Günaydin and Hieronymi, we obtain a full understanding of definable sets in (M, P).