A correspondence principle for the Gowers norms

Informally speaking, the Furstenberg Correspondence [5, 6] shows that the “local behavior” of a dynamical system is controlled by the behavior of sufficiently large finite systems. By the local behavior of a dynamical system (X,B, μ,G), we mean the properties which can be stated using finitely many actions of G and the integral given by μ1. By a finite system, we just mean (S,P(S), c, G) where G is a infinite group, S is a finite quotient of G, and c is the counting measure c(A) := |A| |S| . The most well known example of such a property is the ergodic form of Szemerédi’s Theorem: