Quantitative inverse theorem for Gowers uniformity norms U⁵ and U⁶ in \mathbbF₂ⁿ

2022/07/04 by Luka Milićević, Milićević, Luka
Mathematics · #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Functional Equations Stability Results #Mathematical and Theoretical Analysis #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2207.01591

openalex publication_date 2022/07/04 · openalex created_date 2022/07/07 · openalex updated_date 2026/07/28

Abstract

We prove quantitative bounds for the inverse theorem for Gowers uniformity norms U⁵ and U⁶ in \mathbbF₂ⁿ. 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 U^k norms.

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