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

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


  [1]: https://gdz.sub.uni-goettingen.de/id/PPN365956996_0054?tify=%7B%22view%22%3A%22info%22%2C%22pages%22%3A%5B189%5D%7D