We have provided a model-theoretic proof for the decidability of the additive structure of integers together with the function f mapping x to $$\lfloor \varphi x\rfloor $$ where $$\varphi $$ is the golden ratio.

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. (shrink)