On the distribution of angles between increasingly many short lattice vectors

Following Södergren, we consider a collection of random variables on the space Xn unimodular lattices in dimension n: normalizations angles between N=N(n) shortest vectors lattice, and volumes spheres with radii equal to lengths these vectors. We investigate expected values certain functions (whose support depends parameter K=K(n)) evaluated at regime where K N are allowed tend infinity n rate KN=o(n1/6). Our main result is that as n⟶∞, exhibit joint Poissonian Gaussian behavior.