On the speed of convergence in the ergodic theorem

Given any ergodic invertible measure preserving transformation τ of [0,1] and any null-sequence (α _N ) of positive reals, there exists a continuousf such that $$lim sup alpha _{rm N}^{ - 1} left| {N^{ - 1} sumlimits_{k = 0}^{N - 1} {f circ tau ^k - smallint f} } right| = infty a. e.,$$ i.e. there is no “speed of convergence” in the ergodic theorem for any τ. The analogous result holds also for norm-convergence.