Sean Prendiville

An inverse theorem for the Gowers U^3 norm relative to quadratic level sets

The Gowers uniformity norms have become well-used tools in additive combinatorics, ergodic theory and analytic number theory. We discuss an effective version of the inverse theorem for the Gowers U^3-norm for functions supported on high-rank quadratic level sets in finite vector spaces. This enables one to run density increment arguments with respect to quadratic level sets, which are analogues of Bohr sets in the context of quadratic Fourier analysis on finite vector spaces. For instance, one can derive a polyexponential 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.