高阶Abel代数Toral作用下的数量密度
Quantitative Density under Higher Rank Abelian Algebraic Toral Actions
摘要：
本文将Bourgain-Lindenstrauss-Michel-Venkatesh最近的一维定量密度结果推广到高维环面上的阿贝尔代数作用。在有限指标下，我们所研究的群作用共轭于$U_K$（某个非CM数域$K$的单位群）在$K\otimes_{\mathbb Q}\mathbb R$的紧商上的作用。在这样的设置中，我们调查如何快速的一个泛点的轨道可以成为密集的环面。这effectivizes一个特殊情况下的定理贝伦德;并推导出一个平行的措施理论声明effectivizes一个特殊情况下的结果由Katok-Spatzier。此外，我们指定了两个数值不变量的群作用，确定的定量行为，这具有数论意义。
英文摘要：
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.
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