Anti-uniformity norms, anti-uniformity functions and their algebras on Euclidean spaces

Abstract

Let k ě 2 be an integer. Given a uniform function f -one that satisfies }f } U pkq ă 8, there is an associated anti-uniform function g -one that satisfied }g} Ů pkq . The question is, can one approximate g with the Gowers-Host-Kra dual function D k f of f ? Moreover, given the generalized cubic convolution products D k pfα : α P Ṽk q, what sorts of algebras can they form? In short, this paper explores possible structures of anti-uniformity on Euclidean spaces.