Let be a subvariety of a torus defined over the algebraic numbers. We give a qualitative and quantitative description of the set of points of of height bounded by invariants associated to any variety containing . In particular, we determine whether such a set is or is not dense in . We then prove that these sets can always be written as the intersection of 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 Amoroso and David up to a logarithmic factor
Francesco Amoroso. Evelina Viada. "Small points on subvarieties of a torus." Duke Math. J. 150 (3) 407 - 442, 1 December 2009. https://doi.org/10.1215/00127094-2009-056