Asymptotics for return times of rank-one systems

We give a condition for nonperiodic rank-one systems to have non-exponential asymptotic distribution (equal to 1([1,infinity])) of return times along subsequences of cylinders. Applying this result to the staircase transformation, we derive mixing dynamical systems with non-exponential asymptotics. Moreover, we show for two columns rank-one systems unique asymptotic along full sequences of cylinders.