Slow convergence of Birkhoff ergodic averages

Abstract

Let $\{T^z\}$ be an ergodic action of the group $Z^n$ by automorphisms of the probability space $(X,m)$, $\sum_{i}^\infty a_i<\infty$, $a_i>0$. For any sequence $M_k\to +\infty$ there exist $N_k>M_k$ and a function $ f\in L_1(X,m)$ such that $$m\left(\ x:\ \left|\, A(x,N_k,f) - \int f \, dm\, \right|\ >\ a_k \ \right)\ \to\ 1,$$ where $$A(x,N,f) =\frac 1 {N^n} \sum_{z\in Q_N} f(T^{z}x),$$ $$Q_N=\{(z_1,\dots,z_n)\}\, :\, 1\leq z_1,\dots,z_n\leq N\}.$$