Multiple Recurrence Theorem
Let be a probability space, and let be measure-preserving transformations, for between and . Assume that all the transformations commute. If is a positive measure set , then, there exists such that
In other words there exist a certain time such that the subset of for which all elements return to simultaneously for all transformations is a subset of with positive measure. Observe that the theorem may be applied again to the set , obtaining the existence of such that
So we may conclude that, when has positive measure, there are infinite times for which there is a simultaneous return for a subset of with positive measure.
As a corollary, since the powers of a transformation commute, we have that, for with positive measure there exists such that
As a consequence of the multiple recurrence theorem one may prove Szemerédi’s Theorem about arithmetic progressions.