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

We study the asymptotic distribution of norm ball averages along orbits of a lattice Γ ⊂ SO(n,1) acting on the moduli space of pairs of orthogonal discrete subgroups of Rⁿ⁺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.