诣零流形上稀疏序列的等分布
Equidistribution of sparse sequences on nilmanifolds
摘要：
研究了当参数$n$限制在某个稀疏序列的范围内时，零轨道$(b^nx)_{n\in\N}$的等分布性质.例如，我们证明了，如果$X=G/Γ$是一个nilmanifold，$b\in G$是一个遍历nilrotation，和$c\in \R\setminus \Z$是积极的，那么序列$(b^{[n^c]}x)_{n\in\N}$是均匀分布在$X$的每一个$x\in X$。当用$a(n)$替换$n^c$时，以及当用具有次指数增长的随机整数序列替换$n^c$时，也是这种情况，其中$a(t)$是属于某个哈代域的函数，具有多项式增长，并且在数学上远离多项式。当$X$是圆时，Boshernitzan已经建立了类似的结果。
英文摘要：
We study equidistribution properties of nil-orbits $(b^nx)_{n\in\N}$ when the parameter $n$ is restricted to the range of some sparse sequence that is not necessarily polynomial. For example, we show that if $X=G/Γ$ is a nilmanifold, $b\in G$ is an ergodic nilrotation, and $c\in \R\setminus \Z$ is positive, then the sequence $(b^{[n^c]}x)_{n\in\N}$ is equidistributed in $X$ for every $x\in X$. This is also the case when $n^c$ is replaced with $a(n)$, where $a(t)$ is a function that belongs to some Hardy field, has polynomial growth, and stays logarithmically away from polynomials, and when it is replaced with a random sequence of integers with sub-exponential growth. Similar results have been established by Boshernitzan when $X$ is the circle.
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