A Spectral Criterion for Power-Law Convergence Rate in the Ergodic Theorem for $ {��}^{d} $ and $ {��}^{d} $ Actions

We prove the equivalence of the power-law convergence rate in the \( L_{2} \)-norm of ergodic averages for \( {𝕑}^{d} \) and \( {𝕉}^{d} \) 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 \( d=1 \).