Under recurrence in the Khintchine recurrence theorem

TL;DR

This paper investigates the existence of under-recurrent sets in measure-preserving systems, showing they exist in positive entropy ergodic systems but not in all mixing systems, and explores related recurrence phenomena and combinatorial implications.

Contribution

It demonstrates the presence of under-recurrent sets in positive entropy ergodic systems and clarifies their rarity compared to over-recurrence, answering open questions.

Findings

Positive entropy ergodic systems have under-recurrent sets.

Not all mixing systems possess under-recurrent sets.

Under-recurrence is rarer than over-recurrence.

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.