Spectral algorithms in higher-order Fourier analysis

TL;DR

This paper introduces spectral algorithms for higher-order Fourier analysis, providing a unified framework and new theorems that deepen understanding of Gowers norms and quadratic Fourier structures.

Contribution

It presents a general spectral framework for higher-order Fourier algorithms and establishes new inverse and regularity theorems based on spectral decompositions.

Findings

New spectral inverse theorem for quadratic Fourier analysis

Spectral regularity theorem for Gowers norms

Deeper spectral understanding of higher-order Fourier structures

Abstract

Our goal is to provide simple and practical algorithms in higher-order Fourier analysis which are based on spectral decompositions of operators. We propose a general framework for such algorithms and provide a detailed analysis of the quadratic case. Our results reveal new spectral aspects of the theory underlying higher-order Fourier analysis. Along these lines, we prove new inverse and regularity theorems for the Gowers norms based on higher-order character decompositions. Using these results, we prove a spectral inverse theorem and a spectral regularity theorem in quadratic Fourier analysis.