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} $ -normof ergodic averages for $ {𝕑}^{d} $ and $ {𝕉}^{d} $ actions and the samepower-law estimate for the spectral measure of symmetric $ d $ -dimensionalparallelepipeds: for the degrees that are roots of some special symmetricpolynomial in  $ d $ variables. Particularly, all possible rangeof power-law rates is covered for $ d=1 $ .