Theories of real addition with and without a predicate for integers

We show that it is decidable whether or not a relation on the reals definable in the structure $⟨\mathbb{R},+,<,\mathbb{Z}⟩$ can be defined in the structure $⟨\mathbb{R},+,<,1⟩$. This result is achieved by obtaining a topological characterization of $\langle $\mathbb{R}$, +,<, 1 \rangle$-definablerelationsinthefamilyof$\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 $⟨\mathbb{R},+,<,1,X⟩$. The above characterization allows us to prove that there is no intermediate structure between $⟨\mathbb{R},+,<,\mathbb{Z}⟩$ and $\langle $\mathbb{R}$, +,<, 1 \rangle$\mathrm{.}Wealsoshowthata$\langle $\mathbb{R}$, +,<, $\mathbb{Z}$ \rangle$-definablerelationis$\langle $\mathbb{R}$, +,<, 1 \rangle$-definableifandonlyifitsintersectionwithevery$\langle $\mathbb{R}$, +,<, 1 \rangle$-definablelineis$\langle $\mathbb{R}$, +,<, 1 \rangle$-definable\mathrm{.}Thisgivesanoneffectivebutsimplecharacterizationof$\langle $\mathbb{R}$, +,<, 1 \rangle$-definable relations.