Slowly decaying averages and fat towers

Let $(X,\Sigma,m,\tau)$ be an ergodic system, that is, $(X, \Sigma, m)$ is a probability space and $\tau: X \to X$ is an invertible ergodic $m$-preserving transformation. For a function $f:X\to\mathbb R$, let $A_Nf$ denote the $N$th ergodic average, $A_Nf(x)=\frac{1}{N}\cdot (f(x)+\dots+\tau^ {N-1}f(x))$. Martin Barlow (personal communication) asked the following question, which arose from the work of a student (Zichun Ye) on interface models. Question: If $f(x) \ge 0$ is integrable, and $N(x) = \min \{n: A_kf(x) \le 2 \int f \text{for all} k \ge n\}$, is it the case that $N(x)$ is also integrable? In this note we show that the answer to this Question is no in general, even for bounded functions. In so doing we discover that every ergodic system has a special sort of Kakutani tower which we call a fat tower.