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 $$m\left(x:\left|A(x,N_k,f)-\int fdm\right|>a_k\right)\to 1,$$ where $$A(x,N,f)=\frac{1}{N^n}\sum _{z\in Q_N}f(T^{z}x),$$ $$Q_N=\{(z_1,\dots ,z_n)\}:1\le z_1,\dots ,z_n\le N\}.$$

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