# Multiple Recurrence Theorem

Let $(X,\mathcal{B},\mu)$ be a probability space, and let $T_{i}:X\rightarrow X$ be measure-preserving transformations, for $i$ between $1$ and $q$. Assume that all the transformations $T_{i}$ commute. If $E\subset X$ is a positive measure set $\mu(E)>0$, then, there exists $n\in\mathbb{N}$ such that

$\mu(E\cap T_{1}^{-n}(E)\cap\cdots\cap T_{q}^{-n}(E))>0$ |

In other words there exist a certain time $n$ such that the subset of $E$ for which all elements return to $E$ simultaneously for all transformations $T_{i}$ is a subset of $E$ with positive measure. Observe that the theorem may be applied again to the set $G=E\cap T_{1}^{-n}(E)\cap\cdots\cap T_{q}^{-n}(E)$, obtaining the existence of $m\in\mathbb{N}$ such that

$\mu(G\cap T_{1}^{-m}(G)\cap\cdots\cap T_{q}^{-m}(G))>0$ |

so that

$\mu(E\cap T_{1}^{-(m+n)}(E)\cap\cdots\cap T_{q}^{-(m+n)}(E))\geq\mu(G\cap T_{1% }^{-m}(G)\cap\cdots\cap T_{q}^{-m}(G))>0$ |

So we may conclude that, when $E$ has positive measure, there are infinite times for which there is a simultaneous return for a subset of $E$ with positive measure.

As a corollary, since the powers $T,T^{2}\cdots T^{q}$ of a transformation $T$ commute, we have that, for $E$ with positive measure there exists $n\in\mathbb{N}$ such that

$\mu(E\cap T^{-n}(E)\cap\cdots\cap T^{-qn}(E))>0$ |

As a consequence of the multiple recurrence theorem one may prove Szemerédi’s Theorem about arithmetic progressions.