论文标题
在越来越多的短晶格向量之间的角度分布
On The Distribution Of Angles Between Increasingly Many Short Lattice Vectors
论文作者
论文摘要
在Södergren之后，我们考虑在尺寸$ n $的空间上的一个随机变量集合：$ n = n = n（n）$最短的载体之间的角度正常化的正常化，以随机的单型晶格，以及与这些vectors的长度等于radii的球体量。我们研究了在这些随机变量中评估的某些函数的预期值，在这些随机变量中，$ n $以$ n $以$ n = o \ left（n^{1/6} \ right）$的$ n $倾向于无限。我们的主要结果是，作为$ n \ longrightarrow \ infty $，这些随机变量表现出联合泊松和高斯行为。
Following Södergren, 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.
