Does anyone know if there is a classification of the subgroups of the real numbers taken under addition? If not can anyone point me in the directiong of any papers/materials which discuss properties of or interesting facts about these subgroups? 



Additive Subgroups of the Reals.
If you would like to classify the subgroups in the sense of Lebesugue measure, you may find the following facts helpful.


(1) Any measurable proper subgroup of the real line is of measure $0$.


(2) Any non-measurable subgroup $G$ of the real line charges fully everywhere, i.e., for any interval $I$, $m^{\ast}(G \cap I)=|I|$, where $m^{\ast}(\cdot)$ denotes the outer Lebesgue measure.


(3) Non-measurable subgroup of the real line exists.


