The model theory of the field of reals with a subgroup of the unit circle

We describe definable sets in the field of reals augmented bya predicate for a finite rank multiplicative group of complexnumbers contained in the unit circle . This structure interpretsthe quotient-space / which, for infinite cyclic, is relatedto the quantum torus. Every definable set is proved to be aBoolean combination of existentially definable sets. We givea complete set of axioms for the theory of such a structure.