In "The field of reals with a predicate for the powers of two", Van den Dries has proved that the set of integers is not definable in  $(\mathbb{R}, +,., \leq, 0, 1, 2^{\mathbb{Z}})$, where
  $2^{\mathbb{Z}}=\{2^n: n \in \mathbb{Z} \}$.

**Question.** Is there a subset $S$ of $2^{\mathbb{Z}}$ sucu that $\mathbb{Z}$ is definable in $(\mathbb{R}, +,., \leq, 0, 1, S)?$