# Multiple Recurrence Theorem

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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.
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