Two of the most fundamental tools in ergodic Ramsey theory are the mean ergodic theorem and the van der Corput trick. Both have a classical and fairly simple proof, which I have presented before in the blog. Recently I came across alternative proofs for both results which seem to “not use anything” other than very well known general facts (such as Banach-Alaoglu theorem). Although both facts are related to Cesàro limits along Følner sequences in arbitrary amenable groups, the proofs presented in this post are not related. I don’t think either of the proofs is new, but I don’t remember ever seing them written out; so I decided to post them here.
After writing down the alternative proof of the van der Corput trick, I realized one actually needs somewhat advanced results to obtain the full generality… (However if we care only about , or even abelian groups (or the weak version with only one average), then the proof is softer.) Continue reading