Under recurrence in the Khintchine recurrence theorem

Abstract: The Khintchine recurrence theorem asserts that on a measure preserving system, for every set $A$ and $\varepsilon>0$, we have $\mu(A\cap T^{-n}A)\geq \mu(A)^2-\varepsilon$ for infinitely many $n\in \mathbb{N}$. We show that there are systems having under-recurrent sets $A$, in the sense that the inequality $\mu(A\cap T^{-n}A)< \mu(A)^2$ holds for every $n\in \mathbb{N}$. 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.