Structure Theorems in Tame Expansions of O-minimal Structures by a Dense Set

We study sets and groups definable in tame expansions of ominimal structures. Let M̃ = ⟨M, P ⟩ be an expansion of an o-minimal Lstructure M by a dense set P . We impose three tameness conditions on M̃ and prove a cone decomposition theorem for definable sets and functions in the realm of the o-minimal semi-bounded structures. The proof involves induction on the notion of ‘large dimension’ for definable sets, an invariant which we herewith introduce and analyze. As a corollary, we obtain that (i) the large dimension of a definable set coincides with the combinatorial scl-dimension coming from a pregeometry given in Berenstein-Ealy-Günaydin [3], and (ii) the large dimension is invariant under definable bijections. We then illustrate how our results open the way to the study of groups definable in M̃, by proving that around scl-generic elements of a definable group, the group operation is given by an L-definable map.