标题： 关于越来越多的短格向量之间角度的分布 显示英文标题

标题： On The Distribution Of Angles Between Increasingly Many Short Lattice Vectors

摘要： 按照Södergren的方法，我们考虑在维数为$n$的单模格空间$X_n$上的随机变量集合：随机单模格中$N = N(n)$个最短向量之间的角度的归一化值，以及这些向量长度相等的球体的体积。 我们在$N$随着$n$以$N = o \left( n^{1/6} \right)$的速度趋于无穷大的情况下，研究这些随机变量在某些函数上的期望值。 我们的主要结果是，当$n \longrightarrow \infty$时，这些随机变量表现出联合泊松和高斯行为。 显示英文摘要

摘要： Following S\"odergren, we consider a collection of random variables on the space $X_n$ of unimodular lattices in dimension $n$: Normalizations of the angles between the $N = N(n)$ shortest vectors in a random unimodular lattice, and the volumes of spheres with radii equal to the lengths of these vectors. We investigate the expected values of certain functions evaluated at these random variables in the regime where $N$ tends to infinity with $n$ at the rate $N = o \left( n^{1/6} \right)$. Our main result is that as $n \longrightarrow \infty$, these random variables exhibit a joint Poissonian and Gaussian behaviour.