Distribution of shapes of orthogonal lattices

It was recently shown by Aka, Einsiedler and Shapira that if $d>2$ , the set of primitive vectors on large spheres when projected to the $(d-1)$ -dimensional sphere coupled with the shape of the lattice in their orthogonal complement equidistribute in the product space of the sphere with the space of shapes of $(d-1)$ -dimensional lattices. Specifically, for $d=3,4,5$ some congruence conditions are assumed. By using recent advances in the theory of unipotent flows, we effectivize the dynamical proof to remove those conditions for $d=4,5$ . It also follows that equidistribution takes place with a polynomial error term with respect to the length of the primitive points.