Equidistribution of Iwasawa components of lattices and asymptotic properties of primitive points

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 grids in their orthogonal hyperplanes, and the shortest solutions to their associated gcd equations. These equidistribution statements follow from counting lattice points in the real Special Linear group.