Linear forms and quadratic uniformity for functions on $\mathbb{Z}_N$

TL;DR

This paper advances understanding of uniformity norms in additive combinatorics by proving a key case of a conjecture for functions on cyclic groups of prime order, addressing complexities not present in vector spaces.

Contribution

It proves the first non-trivial case of the main conjecture relating to quadratic uniformity norms on cyclic groups of prime order, extending prior results from vector spaces to these groups.

Findings

Established the first non-trivial case of the main conjecture for quadratic uniformity norms in inite cyclic groups.

Developed techniques to handle local inverse theorems in inite cyclic groups.

Addressed complexities arising from the lack of rich subgroup structure in inite cyclic groups.

Abstract

A very useful fact in additive combinatorics is that analytic expressions that can be used to count the number of structures of various kinds in subsets of Abelian groups are robust under quasirandom perturbations, and moreover that quasirandomness can often be measured by means of certain easily described norms, known as uniformity norms. However, determining which uniformity norms work for which structures turns out to be a surprisingly hard question. In [GW09a] and [GW09b, GW09c] we gave a complete answer to this question for groups of the form $G=\mathbb{F}_p^n$, provided $p$ is not too small. In $\mathbb{Z}_N$, substantial extra difficulties arise, of which the most important is that an "inverse theorem" even for the uniformity norm $\|.\|_{U^3}$ requires a more sophisticated (local) formulation. When $N$ is prime, $\mathbb{Z}_N$ is not rich in subgroups, so one must use regular Bohr neighbourhoods instead. In this paper, we prove the first non-trivial case of the main conjecture from [GW09a].