A Spectral Criterion for Power-Law Convergence Rate in the Ergodic Theorem for  and  Actions.

We prove the equivalence of the power-law convergence rate in the -norm of ergodic averages for and actions and the same power-law estimate for the spectral measure of symmetric -dimensional parallelepipeds: for the degrees that are roots of some special symmetric polynomial in variables. Particularly, all possible range of power-law rates is covered for . [ABSTRACT FROM AUTHOR]