The Khintchine recurrence theorem asserts that on a measure preserving
system, for every set A and ε>0, we have μ(A∩T−nA)≥μ(A)2−ε for infinitely many n∈N. We show that there
are systems having under-recurrent sets A, in the sense that the inequality
μ(A∩T−nA)<μ(A)2 holds for every n∈N. In
particular, all ergodic systems of positive entropy have under-recurrent sets.
On the other hand, answering a question of V.~Bergelson, we show that not all
mixing systems have under-recurrent sets. We also study variants of these
problems where the previous strict inequality is reversed, and deduce that
under-recurrence is a much more rare phenomenon than over-recurrence. Finally,
we study related problems pertaining to multiple recurrence and derive some
interesting combinatorial consequences.Comment: 18 pages. Referee's comments incorporated. To appear in the Israel
Journal of Mathematic