  We consider toral extensions of hyperbolic dynamical systems. We prove that
its quantitative recurrence (also with respect to given observables) and
hitting time scale behavior depend on the arithmetical properties of the
extension. By this we show that those systems have a polynomial decay of
correlations with respect to $C^{r}$ observables, and give estimations for its
exponent, which depend on $r$ and on the arithmetical properties of the system.
We also show examples of systems of this kind having not the shrinking target
property, and having a trivial limit distribution of return time statistics.
