Decomposition of multicorrelation sequences and joint ergodicity

Abstract

We show that, under finitely many ergodicity assumptions, any multicorrelation sequence defined by invertible measure-preserving Z(d) -actions with multivariable integer polynomial iterates is the sum of a nilsequence and a nullsequence, extending a recent result of the second author. To this end, we develop a new seminorm bound estimate for multiple averages by improving the results in a previous work of the first, third, and fourth authors. We also use this approach to obtain new criteria for joint ergodicity of multiple averages with multivariable polynomial iterates on Z(d) -systems.