Small points on subvarieties of a torus

Let V be a subvariety of a torus defined over the algebraic numbers. We give a qualitative and quantitative description of the set of points of V of height bounded by invariants associated to any variety containing V . Especially, we determine whether such a set is or is not dense in V . We then prove that these sets can always be written as the intersection of V with a finite union of translates of tori of which we control the sum of the degrees. As a consequence, we prove a conjecture by the first author and David up to a logarithmic factor.