ATTENTION : EXCEPTIONNELLEMENT, CET EXPOSÉ AURA LIEU À L’INSTITUT HENRI POINCARÉ.

We work in an expansion (M, P) of an o-minimal structure M by a dense set P, such that three tameness conditions hold. Examples include dense pairs, expansions of M by an independent set, and expansions by a multiplicative group with the Mann property. In the first part of the talk, we give a structure theorem for definable sets in (M, P), in analogy with the cell decomposition theorem known for o-minimal structures, and analyze the relevant notion of dimension (joint with Günaydın and Hieronymi). In the second part, we propose a generalized "Pila-Wilkie" statement and prove it in the three examples. Namely, if X is a definable set in (M, P) and contains many rational points, then it is dense in an infinite semialgebraic set. The proof of the statement is by reduction to the standard Pila-Wilkie theorem, using the structure theorem.