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 $K\otimes _\mathbb Q\mathbb R$. 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.