Quantitative inverse theorem for Gowers uniformity norms \(\mathsf{U}^5\) and \(\mathsf{U}^6\) in \(\mathbb{F}_2^n\) (Q6622787)

In this paper, the author establish quantitative bounds for the inverse theorem related to the Gowers uniformity norms \( U_5 \) and \( U_6 \) over the finite field \( \mathbb{F}_2^n \). This work builds on an earlier result by Gowers and the author, which reduced the inverse problem to an investigation of certain algebraic properties of multilinear forms. A central focus here is the action of the symmetric groups \( \text{Sym}_4 \) and \( \text{Sym}_5 \) on the space of these forms, and the authors study how these actions influence the partition rank, using a refined algebraic regularity method to establish effective quantitative bounds.\N\NThese results follow a very fruitful are of research in the interplay between number theory, combinatorics and analysis. The study of inverse theorems for Gowers norms has primarily focused on two types of groups: cyclic groups of prime order, denoted \( \mathbb{Z}/N\mathbb{Z} \), and finite-dimensional vector spaces over prime fields, denoted \( \mathbb{F}_p^n \). When \( G = \mathbb{F}_p^n \) and in the high-characteristic case \( p \geq k \), Bergelson, Tao, and Ziegler [\textit{V. Bergelson} et al., Geom. Funct. Anal. 19, No. 6, 1539--1596 (2010; Zbl 1189.37007)] established an inverse theorem using phases of polynomials as obstructions. Later, \textit{T. Tao} and \textit{T. Ziegler} [Ann. Comb. 16, No. 1, 121--188 (2012; Zbl 1306.11015)] extended this result to the low-characteristic case \( p < k \), employing nonclassical polynomials as the obstruction family. These nonclassical polynomials arise as solutions to extremal problems involving functions \( f \colon \mathbb{F}_p^n \to \mathbb{D} \) with \( \| f \|_{U_k} = 1 \).\N\NIn the case \( G = \mathbb{Z}/N\mathbb{Z} \), Green, Tao, and Ziegler \N[\textit{B. Green} et al., Ann. Math. (2) 176, No. 2, 1231--1372 (2012; Zbl 1282.11007)] proved an inverse theorem that uses nilsequences to characterize obstructions. An alternative framework based on nilspaces was subsequently developed by Szegedy, and further refined by Camarena, Szegedy, and others. Recent work by Candela, González-Sánchez, and Szegedy [\textit{P. Candela} et al., Ergodic Theory Dyn. Syst. 43, No. 12, 3971--4040 (2023; Zbl 1533.37020)] provided an alternative proof of the Tao-Ziegler inverse theorem using nilspace techniques.\N\NRegarding quantitative bounds, previous results have largely relied on infinitary methods and regularity lemmas, leading to ineffective bounds. Green and Tao provided the first effective bounds for the \( U_3 \) norm in abelian groups of odd order, while Samorodnitsky obtained bounds for \( G = \mathbb{F}_2^n \), with a more recent contribution by \textit{A. Jamneshan} and \textit{T. Tao} [``The inverse theorem for the $U^3$ Gowers uniformity norm on arbitrary finite abelian groups: Fourier-analytic and ergodic approaches'', Preprint, \url{arXiv:2112.13759}]. For the \( U_4 \) norm in the vector space setting \( G = \mathbb{F}_p^n \), quantitative bounds were achieved by Gowers and the author when \( p \geq 5 \), and by Tidor for \( p < 5 \). More generally, Manners provided quantitative bounds for arbitrary \( k \) in the \( \mathbb{Z}/N\mathbb{Z} \) setting, while Gowers and the author achieved similar results in \( \mathbb{F}_p^n \) in the high-characteristic case.\N\NA notable outcome of this work is a resolution of Tidor's conjecture on approximately symmetric multilinear forms in five variables, which had been previously disproven for four variables. Finally, the authors discuss potential extensions of their approach to general \( U_k \) norms, exploring both theoretical challenges and possible avenues for further development.