Is there a classification of all countable subgroups of the circle $\mathbb{T} \simeq \mathbb{R}/\mathbb{Z}$? 


It seems that there are quite a lot of them, e.g.:


* cyclic subgroups $\{a^n\colon n\in\mathbb{Z}\}$
* finite subgroups
* subgroups of the form $\{k/2^n\colon k,n\in \mathbb{Z}\}$ or something similar
* direct sums of the above...


Equivalently, is there a classification of all compact monothetic groups? (Each countable subgroup of the circle can be realized as a group of eigenvalues of some probability-preserving transformation and vice versa).



Classification of countable subgroups of the circle
"subgroups of the form $\{k/2n:k,n∈\mathbb{Z}\}$ or something similar [and] direct sums of the above..."


Are precisely the class of divisible subgroups. See <http://en.wikipedia.org/wiki/Divisible_group> for a full discussion and classification. Divisible groups are well-understood, and are always direct summands. As $\mathbb{R}/\mathbb{Z}$ is divisible, we can say a lot about its structure.


$\mathbb{Q}/\mathbb{Z}$ will be the torsion subgroup, which is the direct sum of all prufer groups. This contains all subgroups of the last three types which you list. 


As $\mathbb{Q}/\mathbb{Z}$ is divisible, $\mathbb{R}/\mathbb{Z}$ is isomorphic to $\mathbb{R}/\mathbb{Q}\oplus \mathbb{Q}/\mathbb{Z}$. 


But $\mathbb{R}/\mathbb{Q}$ is just a $\mathbb{Q}$ vector space. So it is a direct sum of copies of $\mathbb{Q}$.


This doesn't exactly answer your question, but it makes some points.


