On The Distribution Of Angles Between Increasingly Many Short Lattice Vectors

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.