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 (M) over tilde = < M, P > be an expansion of an o-minimal L-structure M by a dense set P. We impose three tameness conditions on (M) over tilde and prove a structure theorem for definable sets and functions in analogy with the cell decomposition theorem known for o-minimal structures. The structure theorem advances the state-of-the-art in all known examples of such (M) over tilde, as it achieves a decomposition of definable sets into unions of `cones', instead of only boolean combinations of them. The proofs involve induction on the notion of `large dimension' for definable sets, an invariant which we herewith introduce and analyze. Applications of the cone decomposition theorem include: (i) the large dimension of a definable set coincides with a suitable pregeometric dimension, and it is invariant under definable bijections, (ii) every definable map is given by an L-definable map off a subset of the domain of smaller large dimension, and (iii) around generic elements of a definable group, the group operation is given by an L-definable map.