  Andr\'as Bir\'o and Vera S\'os prove that for any subgroup $G$ of $\T$
generated freely by finitely many generators there is a sequence $A\subset \N$
such that for all $\beta \in \T$ we have ($\|.\|$ denotes the distance to the
nearest integer) $$\beta\in G \Rightarrow \sum_{n\in A} \| n \beta\| <
\infty,\quad \quad \quad \beta\notin G \Rightarrow \limsup_{n\in A, n \to
\infty} \|n \beta\| > 0. $$ We extend this result to arbitrary countable
subgroups of $\T$. 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 $\T$.
