Integer points on spheres and their orthogonal grids

The set of primitive vectors on large spheres in the euclidean space of dimension d⩾3 equidistribute when projected on the unit sphere. We consider here a refinement of this problem concerning the direction of the vector, together with the shape of the lattice in its orthogonal complement. Using unipotent dynamics, we obtained the desired equidistribution result in dimension d⩾6 and in dimension d=4,5 under a mild congruence condition on the square of the radius. The case of d=3 is considered in a separate paper.