Quantitative Density under Higher Rank Abelian Algebraic Toral Actions

Abstract

We generalize Bourgain-Lindenstrauss-Michel-Venkatesh’s recent one-dimensional quantitative density result to abelian algebraic actions 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 ⊗Q 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. ∗Department of Mathematics, Princeton University, Princeton, NJ 08544, USA; zhirenw@math.princeton.edu