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I’ve just uploaded to the ArXiV my paper “Norm convergence of multiple ergodic averages for commuting transformations“, submitted to
Ergodic Theory and Dynamical Systems. This paper settles in full generality the norm convergence problem for several commuting transformations. Specifically, if is a probability space and are commuting measure-preserving transformations, then for any bounded measurable functions , the multiple average
is convergent in the norm topology (and thus also converges in probability). The corresponding question of pointwise almost everywhere convergence remains open (and quite difficult, in my opinion). My argument also does not readily establish a formula as to what this limit actually is (it really establishes that the sequence is Cauchy in rather than convergent).
The l=1 case of this theorem is the classical mean ergodic theorem (also known as the von Neumann ergodic theorem). The l=2 case was established by Conze and Lesigne. The higher l case was partially resolved by Frantzikinakis and Kra, under the additional hypotheses that all of the transformations , as well as the quotients , are ergodic. The special case was established by Host-Kra (with another proof given subsequently by Ziegler). Another relevant result is the Furstenberg-Katznelson theorem, which asserts among other things when is non-negative and not identically zero, then the inner product of the expression (1) with f has a strictly positive limit inferior as . This latter result also implies Szemerédi’s theorem.
It is also known that the Furstenberg-Katznelson theorem can be proven by hypergraph methods, and in fact my paper also proceeds by a hypergraph-inspired approach, although the language of hypergraphs is not explicitly used in the body of the argument. (In contrast to the work of Host-Kra and Ziegler, no nilsystems appear in the proof.)