Structure theorems in tame expansions of o-minimal structures by a dense set

We study sets and groups definable in tame expansions of o-minimal structures. Let $\mathcal {\widetilde M}= \langle \mathcal M, P\rangle$ be an expansion of an o-minimal $\cal L$-structure $\cal M$ by a dense set $P$. We impose three tameness conditions on $\mathcal {\widetilde M}$ and prove a cone decomposition theorem for definable sets and functions in the realm of o-minimal semi-bounded structures. The proofs involve 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 $\operatorname{scl}$-dimension coming from a pregeometry given in Berenstein-Ealy-G\"unaydin, and (ii) the large dimension is invariant under definable bijections. We then illustrate how our results open the way to study groups definable in $\cal {\widetilde M}$, by proving that around $\operatorname{scl}$-generic elements of a definable group, the group operation is given by an $\mathcal L$-definable map.