Subsets of Euclidean space with nearly maximal Gowers norms

TL;DR

This paper characterizes sets in Euclidean space with nearly maximal Gowers norms, showing they are close to ellipsoids, which are the extremal sets for the Gowers norm among equal measure sets.

Contribution

It establishes a stability result indicating that sets with nearly maximal Gowers norms are close to ellipsoids, extending the understanding of extremal configurations.

Findings

Sets with maximal Gowers norm are ellipsoids.

Sets with nearly maximal Gowers norm are close to ellipsoids.

Provides quantitative stability estimates.

Abstract

A set subset of Euclidean space whose indicator function has maximal Gowers norm, among all sets of equal measure, is an ellipsoid up to Lebesgue null sets. If the indicator function has nearly maximal Gowers norm then the set nearly coincides with an ellipsoid.