Furstenberg's multiple recurrence theorem

Remarks

For $k = 1$ the weaker conclusion is the Poincaré recurrence theorem, and the average form follows from the mean ergodic theorem. For general $k$ the proof goes through a structure theorem. One first reduces to ergodic systems. Every ergodic system is built from a trivial system by a possibly transfinite tower of compact extensions, followed possibly by one weakly mixing extension. The property holds for compact extensions, where orbits come back almost periodically. It also holds for weakly mixing extensions, where the sets $T^{-in}A$ behave as if independent. Then it passes up the tower and to limits. Example: for an irrational rotation and an arc $A$ of length $\ell$, take $n$ with $n\alpha$ within $\ell/(2k)$ of an integer. Then $A \cap T^{-n}A \cap \cdots \cap T^{-kn}A$ contains an arc of length at least $\ell/2$. Example: for a Bernoulli scheme and a cylinder set $A$, independence gives $\mu(A \cap T^{-n}A \cap \cdots \cap T^{-kn}A) = \mu(A)^{k+1}$ once $n$ exceeds the length of the cylinder. The hypothesis $\mu(X) < \infty$ cannot be dropped: the translation $x \mapsto x + 1$ on $\R$ with Lebesgue measure and $A = [0, 1)$ gives $A \cap T^{-n}A = \emptyset$ for every $n \ge 1$.

Going deeper

For a student

Furstenberg proved the theorem in 1977 to give an ergodic-theoretic proof of Szemerédi's theorem. The Furstenberg correspondence principle turns a set of integers of positive upper density into a set $A$ of positive measure in a measure-preserving system. It does so in such a way that a positive value of $\mu(A \cap T^{-n}A \cap \cdots \cap T^{-kn}A)$ yields an arithmetic progression $a, a+n, \dots, a+kn$ in the original set. The method grew into ergodic Ramsey theory, with extensions to several commuting transformations, which gives multidimensional Szemerédi theorems, and to polynomial times.