  We show that every invertible strong mixing transformation on a Lebesgue
space has strictly over-recurrent sets. Also, we give an explicit procedure for
constructing strong mixing transformations with no under-recurrent sets. This
answers both parts of a question of V. Bergelson.
  We define $\epsilon$-over-recurrence and show that given $\epsilon > 0$, any
ergodic measure preserving invertible transformation (including discrete
spectrum) has $\epsilon$-over-recurrent sets of arbitrarily small measure.
Discrete spectrum transformations and rotations do not have over-recurrent
sets, but we construct a weak mixing rigid transformation with strictly
over-recurrent sets.
