2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/68203We solve the following counting problem for measure preserving transformations. For $f\in L_+^1(μ)$, 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.34 pages, 2 figuresDynamical Systems37A05, 28D05, 47A35, 60F99text