Convergence Rates in the Ergodic Theorem for Unitary Actions of Compactly Generated Abelian Groups

For the convergence rates in the ergodic theorem for unitary actions of locally compact compactly generated abelian groups, we obtain a spectral criterion in terms of the singularity of the spectral measure in a neighborhood of the unit character. It is thus shown that estimates for these rates, as for the classical ergodic theorems for actions of the groups $ 𝕑 $ and $ 𝕉 $ studied previously, are necessarily of spectral nature.