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The celebrated Birkhoff Ergodic Theorem asserts that, for an ergodic map, orbits of almost every point equidistributes when sampled at integer times. This result was generalized by Bourgain to many natural sparse subsets of the integers. On the other hand, the behaviour of orbits of \textbf{all} points in a dynamical system is much less understood, especially for sparse subsets of the integers. We generalize a method introduced by A. Venkatesh to tackle this problem in two directions, general $\mathbb{R}^d$ actions instead of flows, and weak mixing, rather than mixing, actions. Along the way, we also establish some basic properties of weak mixing and show weak mixing for the time 1-map of a weak mixing flow. The main examples are equidistribution along $\{(n^{1+\epsilon_i})_{i=1}^d\}$, which is a direct generalisation of Venkatesh's example, and equidistribution along random subsets of $\mathbb{Z}^d$ for horosperical action., Comment: 44 pages, no figures, comments are welcome!

We establish abstract local limit theorems for hitting times and return-times of suitable sequences (A_{l}) of asymptotically rare events in ergodic probability preserving dynamical systems, including versions for tuples of consecutive times and positions of the hits. These results are shown to apply in the setup of Gibbs-Markov systems.

We obtain a Poisson Limit for return times to small sets for product systems. Only one factor is required to be hyperbolic while the second factor is only required to satisfy polynomial deviation bounds for ergodic sums. In particular, the second fact can be either elliptic or parabolic. As an application of our main result, several maps of the form Anosov map $\times$ another map are shown to satisfy a Poisson Limit Theorem at typical points, some even at all points. The methods can be extended to certain types of skew products, including $T,T^{-1}$-maps of high rank., Comment: 37 pages, 0 figures, Comments are welcome!