Ergodic-theoretic approach to k-wise correlations and inverse theorems

Develop an ergodic-theoretic framework for studying k-wise correlations of product-space functions and establish inverse theorems in this setting, analogous to the results proved via analytical methods in the paper.

Background

Many advances in uniformity norms and additive combinatorics were influenced by ergodic theory, suggesting potential for an ergodic perspective on general k-wise correlations and inverse results. Such a viewpoint might yield qualitative or conceptual proofs and new tools, even without quantitative bounds.

An ergodic-theoretic treatment could unify disparate settings and inform structural results akin to those established here for pairwise-connected distributions.