Quantitative inverse theorem for gowers uniformity norms U5 and U6 in Fn2

Abstract

We prove quantitative bounds for the inverse theorem for Gowers uniformity norms U5and U6in Fn2. The proof starts from an earlier partial result of Gowers and the author which reduces the inverse problem to a study of algebraic properties of certain multilinear forms. The bulk of the work in this paper is a study of the relationship between the natural actions of Sym4 and Sym5 on the space of multilinear forms and the partition rank, using an algebraic version of regularity method. Along the way, we give a positive answer to a conjecture of Tidor about approximately symmetric multilinear forms in 5 variables, which is known to be false in the case of 4 variables. Finally, we discuss the possible generalization of the argument for Uk norms.