ACTA issues

A divergence criterion and an elementary proof of the divergence of ergodic averages along special subsequences

Minh Dzung Ha

Acta Sci. Math. (Szeged) 68:3-4(2002), 697-703

Abstract. Consider ${\bf T}=\{z \in {\bf C}:|z|=1\}$, the unit circle with the usual normalized arc-length measure ${\cal L}$. We give a simple sufficient condition (a Divergence Criterion), with a completely self-contained and elementary proof, for the divergence of ergodic averages along subsequences in ${\bf N}$. As an application, we give a very elementary argument of the following result. Let $(n_k)_1^\infty$ be any increasing sequence in ${\bf N}$ with strictly increasing gaps, i.e., $n_{k+1}-n_{k}>n_{k}-n_{k-1}, k\geq 2$. Let $0<\rho<1$ be given. Then there exists an ergodic rotation $\tau \colon {\bf T}\to {\bf T}$ such that for any given $\epsilon >0$, there are infinitely many $f \in L^\infty({\bf T})$ satisfying $$ {\cal L}\Big( \big\{z \in {\bf T}: \overline{\lim}{1 \over l}\sum_{k=1}^{l}f \circ \tau^{10^{n_k}}(z)- \underline{\lim}{1 \over l}\sum_{k=1}^lf \circ\tau^{10^{n_k}}(z) \geq \rho \big\}\Big)\geq 1 -\epsilon.$$

AMS Subject Classification (1991): 28D99, 60F99

Received March 27, 2001. (Registered under 2863/2009.)