Quantitative inverse theory of Gowers uniformity norms

TL;DR

This paper discusses the inverse theory of Gowers uniformity norms, highlighting Manners' 2018 breakthrough that provided the first quantitative bounds and new insights into the structure of functions with large uniformity norms.

Contribution

It presents a high-level overview of Manners' new proof of the inverse theorem, emphasizing its quantitative bounds and conceptual advancements.

Findings

Manners' proof offers the first quantitative bounds for the inverse theorem.

The proof introduces new insights into the structure of functions with large Gowers norms.

The bounds are at worst only doubly exponential, improving previous qualitative results.

Abstract

(This text is a survey written for the Bourbaki seminar on the work of F. Manners.) Gowers uniformity norms are the central objects of higher order Fourier analysis, one of the cornerstones of additive combinatorics, and play an important role in both Gowers' proof of Szemer\'{e}di's theorem and the Green-Tao theorem. The inverse theorem states that if a function has a large uniformity norm, which is a robust combinatorial measure of structure, then it must correlate with a nilsequence, which is a highly structured algebraic object. This was proved in a qualitative sense by Green, Tao, and Ziegler, but with a proof that was incapable of providing reasonable bounds. In 2018 Manners achieved a breakthrough by giving a new proof of the inverse theorem. Not only does this new proof contain a wealth of new insights but it also, for the first time, provides quantitative bounds, that are at worst only doubly exponential. This talk will give a high-level overview of what the inverse theorem says, why it is important, and the new proof of Manners.