论文标题
从指数计数到对相关
From exponential counting to pair correlations
论文作者
论文摘要
我们证明了在成倍增长的离散子集$ \ MATHCAL E $ of $ [0，+\ infty [\，$ endove具有权重函数的元素中的元素的相关性上的抽象结果。假设存在$α\ in \ mathbb r $，$ c，δ> 0 $，这样，如$ t \ to+\ to+\ infty $，$ \ mathcal e $元素的加权数字$ \widetildeΩ（t）$不超过$ t $等于$ c \ c \ c \ c \ c \ c \ c \ c \ c \ c \，t^aine^$ t $ t $。我们证明，$ \ Mathcal e $的未量化元素差异的分布函数是$ t \ mapsto \fracΔ2\，e^{ - | t |} $，并且在$ \widetildeΩ（t）$上的错误术语假设下，这对与多态增长的比例相关。我们将此结果应用于封闭的大地测量学和负弯曲的歧管和度量图中的封闭的大地测量学和常见垂直线的一对相关性。
We prove an abstract result on the correlations of pairs of elements in an exponentially growing discrete subset $\mathcal E$ of $[0,+\infty[\,$ endowed with a weight function. Assume that there exist $α\in\mathbb R$, $c,δ>0$ such that, as $t\to+\infty$, the weighted number $\widetildeω(t)$ of elements of $\mathcal E$ that are not greater than $t$ is equivalent to $c\,t^αe^{δt}$. We prove that the distribution function of the unscaled differences of elements of $\mathcal E$ is $t\mapsto\fracδ2\,e^{-|t|}$, and that, under an error term assumption on $\widetildeω(t)$, the pair correlation with a scaling with polynomial growth exhibits a Poissonian behaviour. We apply this result to answer a question of Pollicott and Sharp on the pair correlations of closed geodesics and common perpendiculars in negatively curved manifolds and metric graphs.
