On Gowers norms of some functions

TL;DR

This paper investigates the relationship between Gowers norms of certain two-dimensional functions and their transformations, introducing methods based on transference principles for sums over linear systems.

Contribution

It establishes a connection between the smallness of Gowers norms of functions and their transformed versions, extending to higher norms and employing a novel transference approach.

Findings

Small Gowers norm of f(x,y) implies small norm of f(x,x+y)

Extended results for higher Gowers norms

Introduces a transference method for sums over linear systems

Abstract

We consider a class of two-dimensional functions f(x,y) with the property that the smallness of its rectangular norm implies the smallness of rectangular norm for f(x,x+y). Also we study a family of functions f(x,y) having a similar property for higher Gowers norms. The method based on a transference principle for a class of sums over special systems of linear equations.