Sean Prendiville: An inverse theorem for the Gowers U^3-norm relative to quadratic level sets

Speaker
Sean Prendiville, Lancaster University

Abstract
Gowers uniformity norms have become well-used tools in the (higher-order) circle method, allowing one to count certain arithmetic configurations of interest – configurations out of reach of the classical circle method.  Central to the utility of these norms are so-called inverse theorems, which characterise when they are large. We discuss an effective inverse theorem for the Gowers U^3-norm, localised to a natural class of associated objects. This enables one to perform the (quadratic) circle method locally, and thereby improve error terms when counting. We discuss the benefit of localising in additive combinatorics, improving a bound on the Ramsey number of three-term progressions which are the same colour as their common difference (“Brauer quadruples”), a result it seems difficult to obtain by other means.