A spectral criterion for power-law convergence rate in the ergodic theorem for ${\Bbb Z}^d$ and ${\Bbb R}^d$ actions

We prove the equivalence of the power-law convergence rate in the $L_2$-norm of ergodic averages for ${\Bbb Z}^d$ and ${\Bbb R}^d$ actions and the same power-law estimate for the spectral measure of symmetric $d$-dimensional parallelepipeds: for the degrees that are roots of some special symmetric polynomial in $d$ variables. Particularly, all possible range of power-law rates is covered for $d=1$.