Averages over integer points of the sphere @Asaf: What you say is not quite correct. There are several similar results that follow from the works of Iwaniec and Duke. Equidistribution of lattice points on the sphere, Heegner points, and equidistribution of closed geodesics. For the equidistribution of closed geodesics analog indeed ELMV give a proof based on Linnik's ideas, but for the problem discussed here GH from MO has described the situation carefully.