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

Abstract

Given a theory T of a polynomially bounded o-minimal expansion R of ${\bar{\mathbb{R}} = \langle\mathbb{R}, +, ., 0, 1, < \rangle}$ with field of exponents ${\mathbb{Q}}$, we introduce a theory ${\mathbb{T}}$ whose models are expansions of dense pairs of models of T by a discrete multiplicative group. We prove that ${\mathbb{T}}$ is complete and admits quantifier elimination when predicates are added for certain existential formulas. In particular, if T = RCF then ${\mathbb{T}}$ axiomatises ${\langle\bar{\mathbb{R}}, \mathbb{R}_{alg}, 2^{\mathbb{Z}}\rangle}$, where ${\mathbb{R}_{alg}}$ denotes the real algebraic numbers. We describe types and definable sets in our models and prove that ${\mathbb{T}}$ is dependent.