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

TL;DR

This paper investigates the structure of anti-uniform functions and their algebras on Euclidean spaces, focusing on approximation by Gowers-Host-Kra dual functions and the algebraic properties of generalized cubic convolution products.

Contribution

It introduces a framework for understanding anti-uniformity norms and functions on Euclidean spaces, and explores their algebraic structures and approximation properties.

Findings

Anti-uniform functions can be approximated by Gowers-Host-Kra dual functions.

Generalized cubic convolution products form specific algebraic structures.

The paper characterizes the algebraic properties of anti-uniformity on Euclidean spaces.

Abstract

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.