Under-recurrence in the Khintchine recurrence theorem

Abstract

The Khintchine recurrence theorem asserts that in a measure preserving system, for every set A and e > 0, we have μ(A ∩ T−nA) ≥ μ(A)2 − e for infinitely many n ∈ N. We show that there are systems having underrecurrent 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.