Description
I will discuss the equidistribution of certain parameters of primitive integral points in Euclidean space, as their norms tend to infinity. These parameters include directions of integral points on the unit sphere, the integral lattices in their orthogonal hyperplanes, and the shortest solutions to their associated gcd equations. Passing from primitive vectors to their orthogonal lattices, these results can also be interpreted as equidistribution properties for (primitive) $(n-1)$-dimensional sublattices of $\mathbb{Z}^n$; following W. G. Schmidt, we generalize these results to primitive sublattices of $\mathbb{Z}^n$ of any dimension.
The key tool is counting lattice points in the Lie group $\mathrm{SL}_n(\mathbb{R})$, via an ergodic theorem.