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

Abstract

We prove 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. For configurations controlled by the $U^3$-norm (complexity-two configurations), this enables one to run a density increment argument with respect to quadratic level sets, which are analogues of Bohr sets in the context of quadratic Fourier analysis on finite vector spaces. We demonstrate such an argument by deriving an exponential bound on the Ramsey number of three-term progressions which are the same colour as their common difference (``Brauer quadruples''), a result we have been unable to establish by other means. Our methods also yield polylogarithmic bounds on the density of sets lacking translation-invariant configurations of complexity two. Such bounds for four-term progressions were obtained by Green and Tao using a simpler weak-regularity argument. In an appendix, we give an example of how to generalise Green and Tao's argument to other translation-invariant configurations of complexity two. However, this crucially relies on an estimate coming from the Croot-Lev-Pach polynomial method, which may not be applicable to all systems of complexity two. Hence running a density increment with respect to quadratic level sets may still prove useful for such problems. It may also serve as a model for running density increments on more general nil-Bohr sets, with a view to effectivising other Szemerédi-type theorems.