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.)
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