Limit distributions for SO (n, 1) action on k-lattices in R^n+1

We study the asymptotic distribution of norm ball averages along orbits of a lattice SO (n, 1) acting on the moduli space of pairs of orthogonal discrete subgroups of R^n+1 up to homothety. Our main result shows that, except for special 2-lattices in R³ lying in hyperplanes tangent to the light cone, these measures converge to an explicit semi-invariant probability measure supported on the space of homothety classes of pairs of orthogonal lattices tangent to the light cone. Our main motivation is a conjecture of Sargent and Shapira, which is resolved as a special case of our general result.