The variance of the number of lattice points inside the dilated bounded set $rD$ with random position in ${ℝ}^d$ has asymptotics $\sim r^{d-1}$ if the rotational average of the squared modulus of the Fourier transform of the set is $O\left({\rho }^{-d-1}\right)$. The asymptotics follow from Wiener’s Tauberian theorem.