Small points on subvarieties of a torus

Abstract

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 $V$. In particular, 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 Amoroso and David up to a logarithmic factor