Convergence of Diagonal Ergodic Averages

The case l = 1 is the mean ergodic theorem, and the result can be viewed as a generalization of that theorem. The l = 2 case was proven by Conze and Lesigne [Conze and Lesigne, 1984], and various special cases for higher l have been shown by Zhang [Zhang, 1996], Frantzikinakis and Kra [Frantzikinakis and Kra, 2005], Lesigne [Lesigne, 1993], and Host and Kra [Host and Kra, 2005]. Tao’s argument is unusual, in that he uses the Furstenberg correspondence principle, which is traditionally used to obtain combinatorial results via ergodic proofs, in reverse: he takes the ergodic system and produces a sequence of finite structures. He then proves a related result for these finitary systems and shows that a counterexample in the ergodic setting would give rise to a counterexample in the finite setting. This paper began as an attempt to translate Tao’s argument into a purely infinite one. The primary obstacle to this, as Tao points out ([Tao, 2007b]), is that the finitary setting provides a product structure which isn’t present in the infinitary setting. In order to reproduce it, we have to go by an indirect route, passing through the finitary setting to produce a more highly structured dynamical system. The structure needed, however, is not the full measure theoretic product. What is needed in the finitary setting is a certain disentanglement of the transformations, which amounts to requiring that the underlying set of points be a product of l sets, with the i-th transformation acting only the i-th coordinate, together with a “nice” projection under a certain canonical factor. We obtain this in the infinitary setting using an argument from non-standard analysis. A measure space with this property gives rise to measure spaces on each coordinate, but need not be the product of these spaces: it could contain additional measurable sets