Quantitative density under higher rank abelian algebraic toral actions

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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.

Original languageEnglish (US)
Pages (from-to)3744-3821
Number of pages78
JournalInternational Mathematics Research Notices
Issue number16
StatePublished - 2011

All Science Journal Classification (ASJC) codes

  • General Mathematics


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