Slow convergence of Birkhoff averages is realized for ergodic Z-actions by a Rokhlin tower construction and claimed, without proof, for Z^n-actions.
Krengel, On the speed of convergence in the ergodic theorem Monatsh
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Slow convergence of Birkhoff ergodic averages
Slow convergence of Birkhoff averages is realized for ergodic Z-actions by a Rokhlin tower construction and claimed, without proof, for Z^n-actions.