  We show that it is decidable whether or not a relation on the reals definable
in the structure $\langle \mathbb{R}, +,<, \mathbb{Z} \rangle$ can be defined
in the structure $\langle \mathbb{R}, +,<, 1 \rangle$. This result is achieved
by obtaining a topological characterization of $\langle \mathbb{R}, +,<, 1
\rangle$-definable relations in the family of $\langle \mathbb{R}, +,<,
\mathbb{Z} \rangle$-definable relations and then by following Muchnik's
approach of showing that the characterization of the relation $X$ can be
expressed in the logic of $\langle \mathbb{R}, +,<,1, X \rangle$.
  The above characterization allows us to prove that there is no intermediate
structure between $\langle \mathbb{R}, +,<, \mathbb{Z} \rangle$ and $\langle
\mathbb{R}, +,<, 1 \rangle$. We also show that a $\langle \mathbb{R}, +,<,
\mathbb{Z} \rangle$-definable relation is $\langle \mathbb{R}, +,<, 1
\rangle$-definable if and only if its intersection with every $\langle
\mathbb{R}, +,<, 1 \rangle$-definable line is $\langle \mathbb{R}, +,<, 1
\rangle$-definable. This gives a noneffective but simple characterization of
$\langle \mathbb{R}, +,<, 1 \rangle$-definable relations.
