Quantitative Density under Higher Rank Abelian Algebraic Toral Actions

2010/04/01 by Zhiren Wang, Wang, Zhiren
Mathematics · #37A45 #Advanced Topology and Set Theory #Algebraic Geometry and Number Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1004.0035

openalex publication_date 2010/04/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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⊗_{\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.

Citations

• Invariant measures for higher-rank hyperbolic abelian actions

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