Expansions of subfields of the real field by a discrete set

Let K be a subfield of the real field, D subset of K be a discrete set and f: D(n) -> K be such that f(D(n)) is somewhere dense. Then (K, f) defines Z. We present several applications of this result. We show that K expanded by predicates for different cyclic multiplicative subgroups defines Z. Moreover, we prove that every definably complete expansion of a subfield of the real field satisfies an analogue of the Baire category theorem.