  Let $\tau_r(x,x_0)$ be the time needed for a point $x$ to enter for the first
time in a ball $B_r(x_0)$ centered in $x_0$, with small radius $r$. We
construct a class of translations on the two torus having particular arithmetic
properties (Liouville components with intertwined denominators of convergents)
not satisfying a logarithm law, i.e. such that for generic $x,x_0$
\liminf_{r\to 0} \frac{\log \tau_r(x,x_0)}{-\log r} = \infty.
  By considering a suitable reparametrization of the flow generated by a
suspension of this translation, using a previous construction by Fayad, we show
the existence of a mixing system on three torus having the same properties. The
speed of mixing of this example must be subpolynomial, because we also show
that: in a system having polynomial decay of correlations the above ratio of
logarithms (which is also called the lower hitting time indicator) is bounded
(it is a function of the local dimension and the speed of correlation decay).
  More generally, this shows that reparametrizations of torus translations
having a Liouville component cannot be polynomially mixing.
