On Sums of Indicator Functions in Dynamical Systems

In this paper, we are interested in the limit theorem question for sums of indicator functions. We show that in every aperiodic dynamical system, for every increasing sequence $(a_n)_{n\in\N}\subset\R_+$ such that $a_n\nearrow\infty$ and $\frac{a_n}{n}\to 0$ as $n\to\infty$, there exists a measurable set $A$ such that the sequence of the distributions of the partial sums $\frac{1}{a_n}\sum_{i=0}^{n-1}(\ind_A-\mu(A))\circ T^i$ is dense in the set of the probability measures on $\R$. Further, in the ergodic case, we prove that there exists a dense $G_\delta$ of such sets.