An analogue of a conjecture of Mazur a question in Diophantine approximation on tori

B. Mazur has considered the question of density in the Euclidean topology of the set of $\mathbb{Q}$-rational points on a variety $X$ defined over $\mathbb{Q}$, in particular for Abelian varieties. In this paper we consider the question of closures of the image of finitely generated subgroups of $T(\mathbb{Q})$ in $\mathrm{\Gamma }\mathrm{\backslash }T(\mathbb{R})$ where $T$ is a torus defined over $\mathbb{Q}$, $\mathrm{\Gamma }$ an arithmetic subgroup such that $\mathrm{\Gamma }\mathrm{\backslash }T(\mathbb{R})$ is compact. Assuming Schanuel’s conjecture, we prove that the closures correspond to sub {\it algebraic} tori of $T$.