An ergodic Szemerédi theorem for commuting transformations

The classical Poincar6 recurrence theorem asserts that under the action of a measure preserving transformation T of a finite measure space (X, ~, p.), every set A of positive measure recurs in the sense that for some n > 0,/z (T-'A n A) > 0. In [1] this was extended to multiple recurrence: the transformations T, T2, ..., T k have a common power satisfying /x (A n T-hA n ... n T-k"A)> 0 for a set A of positive measure. We also showed that this result implies Szemer6di's theorem stating that any set of integers of positive upper density contains arbitrarily long arithmetic progressions. In [2] a topological analogue of this is proved: if T is a homeomorphism of a compact metric space X, for any e >0 and k = 1,2,3,..-, there is a point x E X and a common power of T, T 2, 9 9 9 T k such that d(x, Tnx) < e, d(x, T2"x) < e,. 9 d(x, Tk~x) < e. This (weaker) result, in turn, implies van der Waerden's theorem on arithmetic progressions for partitions of the integers. Now in this case a virtually identical argument shows that the topological result is true for any k commuting transformations. This would lead one to expect that the measure theoretic result is also true for arbitrary commuting transformations. (It is easy to give a counterexample with noncommuting transformations.) We prove this in what follows. A corollary is the multidimensional extension of Szemer6di's theorem: Theorem B. Let S C Z" be a subset with positive upper density and let F C Z" be any finite configuration. Then there exists an integer d and a vector n E Z" such that n+dFCS.