Cubic averages and large intersections (Q2786793)

The authors study phenomena of large limits ``along cubic averages''. The main results concern finite sets of continuous polynomial mappings from a locally compact amenable group to a nilpotent group of measure-preserving mappings, and continuous homomorphisms from a finite set of locally compact amenable groups to a nilpotent group of measure preserving transformations of a measure space. The goal of the paper is to show that the phenomenon of large limits along cubic configurations has a very general scope and is based on a simple ``Fubini principle''. This principle states that the limits of uniform multiparameter Cesàro averages can be replaced by iterated limits of uniform Cesàro averages. The authors obtain a general version of the Furstenberg correspondence principle that applies to uncountable amenable discrete groups. They use it to obtain combinatorial corollaries of ergodic results involving large limits.NEWLINENEWLINEFor the entire collection see [Zbl 1309.37003].