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 $(\mathbb{R}, +,., \leq, 0, 1, S)?$

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 $(\mathbb{R}, +,\cdot, \leq, 0, 1, 2^{\mathbb{Z}})$, 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 $(\mathbb{R}, +,\cdot, \leq, 0, 1, S)?$