Quantitative inverse theorem for Gowers uniformity norms and in

Abstract

Abstract We prove quantitative bounds for the inverse theorem for Gowers uniformity norms $\mathsf {U}^5$ and $\mathsf {U}^6$ in $\mathbb {F}_2^n$ . 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 $\operatorname {Sym}_4$ and $\operatorname {Sym}_5$ 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 five variables, which is known to be false in the case of four variables. Finally, we discuss the possible generalization of the argument for $\mathsf {U}^k$ norms.