Quantitative density under higher rank abelian algebraic toral actions

We generalize Bourgain-Lindenstrauss-Michel-Venkatesh's recent one-dimensional quantitative density result to abelian algebraic actions of rank at least 2 on higher dimensional tori. Up to finite index, the group actions that we study are conjugate to the action of UK, the group of units of some non-CM number field K, on a compact quotient of. In such a setting, we investigate how fast the orbit of a generic point can become dense in the torus. This effectivizes a special case of a theorem of Berend; and is deduced from a parallel measure-theoretical statement which effectivizes a special case of a result by Katok-Spatzier. In addition, we specify two numerical invariants of the group action that determine the quantitative behavior, which have number-theoretical significance.