Expansions of the ordered additive group of real numbers by two discrete subgroups

Abstract

The theory of $(\mathbb{R},<,+,\mathbb{Z},\mathbb{Z}a)$ is decidable if $a$ is quadratic. If $a$ is the golden ratio, $(\mathbb{R},<,+,\mathbb{Z},\mathbb{Z}a)$ defines multiplication by $a$. The results are established by using the Ostrowski numeration system based on the continued fraction expansion of $a$ to define the above structures in monadic second order logic of one successor. The converse that $(\mathbb{R},<,+,\mathbb{Z},\mathbb{Z}a)$ defines monadic second order logic of one successor, will also be established.