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

Let $k\geq 2$ be an integer. Given a uniform function $f$ - one that satisfies $\|f\|_{ U(k)}<\infty$, there is an associated anti-uniform function $g$ - one that satisfied $\|g\|_{ U(k)}^{ *}$. 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.

Categories: math.CA