Asymptotics of variance of the lattice point count

The variance of the number of lattice points inside the dilated bounded set rD with random position in ℝ ^d has asymptotics ∼ r ^d−1 if the rotational average of the squared modulus of the Fourier transform of the set is O(ϰ ^−d−1). The asymptotics follow from Wiener’s Tauberian theorem.