For $\mathcal{M}$ an expansion of $\mathcal{R}=(\mathbb{R};+,\times)$ and $A\subseteq\mathbb{R}$, let $\tau^\mathcal{M}_A$ be the topology on $\mathbb{R}$ generated by the sets definable in $\mathcal{M}$ with parameters from $A$. Say that $\mathcal{M}$ is restrained iff for every $A\subseteq\mathbb{R}$ there is a $B\subseteq\mathbb{R}$ such that $\tau_A^\mathcal{M}=\tau_B^\mathcal{R}$. Given $U\subseteq\mathbb{R}$, let $\mathcal{R}_U=(\mathbb{R};+,\times,U)$.
I'm interested in which sets $U$ yield restrained expansions. For instance, $\mathbb{Z}$ does not but $\{2^z:z\in\mathbb{Z}\}$ does; see this MSE question for more detail on this point. Intuitively, very "sparse" countable sets seem likely to be restrained. To keep things hopefully manageable, let me focus on the following:
What are the countable closed sets $U$ such that $\mathcal{R}_U$ is restrained?
In particular, is there a non-d-minimal countable closed set $U$ such that $\mathcal{R}_U$ is restrained?
