We will consider the behavior of the time which is needed for a typical point to enter in a sequence of decreasing targets. In several systems this time increases (having the same scaling behavior) as the inverse of the measure of the targets. We will see that a general condition for this to happen is superpolynomial decay of correlations. We will also see some applications, on the geometric Lorenz flow and geodesic flows in variable negative curvature. On the other side there are translations of the torus having particular arithmetical properties where the time needed to enter in a given sequence of balls increases much faster than the inverse of the ball’s measure. By a construction of Fayad, moreover we can also show smooth, mixing examples of the same kind.