Strong characterizing sequences of countable groups

Andr'as Bir'o and Vera S'os prove that for any subgroup $G$ of $\text{T}$ generated freely by finitely many generators there is a sequence $A\subset \mathbb{N}$ such that for all $\beta \in \text{T}$ we have ($\mathrm{\parallel }\mathrm{.}\mathrm{\parallel }$ denotes the distance to the nearest integer) $$\beta \in G\Rightarrow \sum _{n\in A}\mathrm{\parallel }n\beta \mathrm{\parallel }<\mathrm{\infty },\beta \mathrm{\notin }G\Rightarrow \underset{n\in A,n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\mathrm{\parallel }n\beta \mathrm{\parallel }>0.$$ We extend this result to arbitrary countable subgroups of $\text{T}$. We also show that not only the sum of norms but the sum of arbitrary small powers of these norms can be kept small. Our proof combines ideas from the above article with new methods, involving a filter characterization of subgroups of $\text{T}$.