论文标题
配对相关问题，并将晶格点计入Zeta函数
A pair correlation problem, and counting lattice points with the zeta function
论文作者
论文摘要
这对相关是单位间隔中序列的局部统计量。相对于此统计数据，伪随机行为称为泊松行为。 $（a_nα）的序列的配对相关性的度量理论_ {n \ geq 1} $由Rudnick，Sarnak和Zaharescu开创。这里$α$是一个真实的参数，$（a_n）_ {n \ geq 1} $是一个整数序列，通常是算术来源。最近，就整数序列$（a_n）_ {n \ geq 1} $而言，几乎每个实际数字$α$几乎每个实际数字$α$的Poissonian对相关性提供了标准。在本文中，当$（a_n）_ {n \ geq 1} $是一系列真实而不是整数时，我们为案例开发了类似的框架，从而追求了Rudnick和Technau最近发起的一系列研究。作为我们方法的应用，我们证明，对于每个实际数字$θ> 1 $，序列$（n^θα）_ {n \ geq 1} $具有几乎所有$α\ in \ mathbb {r} $的poissonian对相关性。
The pair correlation is a localized statistic for sequences in the unit interval. Pseudo-random behavior with respect to this statistic is called Poissonian behavior. The metric theory of pair correlations of sequences of the form $(a_n α)_{n \geq 1}$ has been pioneered by Rudnick, Sarnak and Zaharescu. Here $α$ is a real parameter, and $(a_n)_{n \geq 1}$ is an integer sequence, often of arithmetic origin. Recently, a general framework was developed which gives criteria for Poissonian pair correlation of such sequences for almost every real number $α$, in terms of the additive energy of the integer sequence $(a_n)_{n \geq 1}$. In the present paper we develop a similar framework for the case when $(a_n)_{n \geq 1}$ is a sequence of reals rather than integers, thereby pursuing a line of research which was recently initiated by Rudnick and Technau. As an application of our method, we prove that for every real number $θ>1$, the sequence $(n^θα)_{n \geq 1}$ has Poissonian pair correlation for almost all $α\in \mathbb{R}$.
