A criterion for the power-law rate of convergence of ergodic means for unitary actions of $\mathbb{Z}^d$ and $\mathbb{R}^d$

For averaging over parallelepipeds of unitary actions of the groups $\mathbb{Z}^d$ and $\mathbb{R}^d$, a criterion for the power-law rate of norm convergence is obtained for all possible exponents. The proof is based on the study of the asymptotic behavior of integrals of the product of cardinal sines.