The field of reals with a predicate for the real algebraic numbers and a predicate for the integer powers of two

Given a theory T of a polynomially bounded o-minimal expansion R of with field of exponents , we introduce a theory whose models are expansions of dense pairs of models of T by a discrete multiplicative group. We prove that is complete and admits quantifier elimination when predicates are added for certain existential formulas. In particular, if T = RCF then axiomatises , where denotes the real algebraic numbers. We describe types and definable sets in our models and prove that is dependent.