Multiple Recurrence Theorem

Let $(X,ℬ,\mu )$ be a probability space, and let $T_i:X\to X$ 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

$$\mu (E\cap T_1^{-n}(E)\cap \mathrm{\cdots }\cap T_q^{-n}(E))>0$$

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 $G=E\cap T_1^{-n}(E)\cap \mathrm{\cdots }\cap T_q^{-n}(E)$, obtaining the existence of $m\in \mathbb{N}$ such that

$$\mu (G\cap T_1^{-m}(G)\cap \mathrm{\cdots }\cap T_q^{-m}(G))>0$$

so that

$$\mu (E\cap T_1^{-(m+n)}(E)\cap \mathrm{\cdots }\cap T_q^{-(m+n)}(E))\ge \mu (G\cap T_1^{-m}(G)\cap \mathrm{\cdots }\cap T_q^{-m}(G))>0$$

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

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

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