Remarks on the set of values of the ergodic sums of an integer valued function

For an ergodic measure preserving dynamical system $(X, \cal B, \mu, T)$ and a measurable function $f$ with values in $\mathbb{Z}$, we consider for $x \in X$ the set of values of the ergodic sums $S_nf(x):= \sum_0^{n-1} f(T^k x), n \geq 1$.

If $f$ is integrable with $\mu(f) > 0$, several properties of this set (from the point of view of recurrence or arithmetic sets) are simple consequences of Bourgain’s results (1989).

For example, the set ${S_nf(x), n \geq 1}$ contains infinitely many squares for a.e. $x$. If $f$ is not integrable, this property may fail, as shown by a construction of M. Boshernitzan. We give also a counter-example of an integrable centered function $f$ for which the cocycle $(S_nf(x), n \geq 1)$ is non regular and the property fails.