The elementary theory of Dedekind cuts in polynomially bounded structures

Let M be a polynomially bounded, o-minimal structure with archimedean prime model, for example if M is a real closed field. Let C be a convex and unbounded subset of M . We determine the first order theory of the structure M expanded by the set C . We do this also over any given set of parameters from M , which yields a description of all subsets of M n , definable in the expanded structure.