  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.
