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

Abstract

Let $k\ge 2$ be an integer. Given a uniform function $f$ - one that satisfies $\Vert f{\Vert }_{U(k)}<\infty$, there is an associated anti-uniform function $g$ - one that satisfied $\Vert g{\Vert }_{U(k)}^{\ast }$. 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 $D_k(f_{\alpha }:\alpha \in {\tilde{V}}_k)$, what sorts of algebras can they form? In short, this paper explores possible structures of anti-uniformity on Euclidean spaces.