沿Hardy域函数的幂零流形上的均匀分布
Uniform distribution in nilmanifolds along functions from a Hardy field
AI总结：
该研究将幂零流形平移序列的等分布准则及轨道闭包结论从多项式情形推广到Hardy域中的多项式增长函数类，拓展了已有相关研究的适用范围。
摘要：
我们研究了幂零流形上沿Hardy域中多项式增长函数的平移的等分布性质。更确切地说，若$X=G/Γ$是一个幂零流形，$a_1,\ldots,a_k\in G$是可交换的幂零旋转，$f_1,\ldots,f_k$是来自Hardy域的多项式增长函数，我们证明：其一，序列$a_1^{f_1(n)}\cdot\ldots\cdot a_k^{f_k(n)}Γ$的分布由其在极大因子环面上的投影决定，这将Leibman等分布准则从多项式推广到了更广泛的函数类；其二，$a_1^{f_1(n)}\cdot\ldots\cdot a_k^{f_k(n)}Γ$的轨道闭包始终是子幂零流形的有限并，拓展了Leibman与Frantzikinakis此前在该主题上的部分研究成果。
英文摘要：
We study equidistribution properties of translations on nilmanifolds along functions of polynomial growth from a Hardy field. More precisely, if $X=G/Γ$ is a nilmanifold, $a_1,\ldots,a_k\in G$ are commuting nilrotations, and $f_1,\ldots,f_k$ are functions of polynomial growth from a Hardy field then we show that $\bullet$ the distribution of the sequence $a_1^{f_1(n)}\cdot\ldots\cdot a_k^{f_k(n)}Γ$ is governed by its projection onto the maximal factor torus, which extends Leibman's Equidistribution Criterion form polynomials to a much wider range of functions; and $\bullet$ the orbit closure of $a_1^{f_1(n)}\cdot\ldots\cdot a_k^{f_k(n)}Γ$ is always a finite union of sub-nilmanifolds, which extends some of the previous work of Leibman and Frantzikinakis on this topic.
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