Quantitative Density under Higher Rank Abelian Algebraic Toral Actions

TL;DR

This paper extends density results for higher-dimensional toral actions by abelian algebraic groups, providing effective bounds and linking the dynamics to number-theoretic invariants.

Contribution

It generalizes one-dimensional density results to higher dimensions, introduces numerical invariants, and connects dynamical behavior with algebraic number theory.

Findings

Orbit density rates depend on specific number-theoretic invariants.

Provides effective bounds for the speed of density in higher-dimensional tori.

Links dynamical properties to algebraic structures of number fields.

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 $U_K$, 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.