  We solve the following counting problem for measure preserving
transformations. For $f\in L_+^1(\mu)$, is it true that $\ds
\sup_n\frac{\bN_n(f)(x)}{n} <\infty,$ where $$\ds\bN_n(f)(x)= # {k: \frac{f(T^k
x)}{k}>\frac 1 n}?$$ One of the consequences is the nonvalidity of J.
Bourgain's Return Time Theorem for pairs of $(L^1, L^1)$ functions.

Dynamical Systems