---
tag: E459H2B2EM
kind: statement
role: theorem
name: Furstenberg's multiple recurrence theorem
msc: [37A05, 37A30, 11B25]
uses: [0NFPJVJJHX, 556WBDAHNP, TVYJPBHD0S, B23B0QEMN5]
about: [556WBDAHNP]
sources:
  - url: https://en.wikipedia.org/wiki/Szemer%C3%A9di%27s_theorem
---

## Statement

Let $(X, \mathcal{B}, \mu)$ be a [[probability space|0NFPJVJJHX]] and let $T \colon X \to X$ be a [[measure-preserving transformation|556WBDAHNP]].
Let $A \in \mathcal{B}$ with $\mu(A) > 0$, and let $k \ge 1$ be an integer.
Then
$$\liminf_{N \to \infty} \frac{1}{N} \sum_{n=1}^{N} \mu\big(A \cap T^{-n}A \cap T^{-2n}A \cap \cdots \cap T^{-kn}A\big) > 0.$$
In particular, there is an integer $n \ge 1$ with $\mu(A \cap T^{-n}A \cap \cdots \cap T^{-kn}A) > 0$.

## Remarks

For $k = 1$ the weaker conclusion is the [[Poincaré recurrence theorem|TVYJPBHD0S]], 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 $n \ge 1$.

## Explanations

### L1 human

In a system that preserves a finite volume, a set of positive volume does not only come back to itself.
It comes back at evenly spaced times $n, 2n, \dots, kn$ simultaneously, for some $n$ and for any length $k$ you ask for.
This is a dynamical form of finding long arithmetic progressions.

### L2 human

Furstenberg proved the theorem in 1977 to give an ergodic-theoretic proof of [[Szemerédi's theorem|B23B0QEMN5]].
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.