论文标题
Gowers统一性规范的定量逆定理$ \ MATHSF {u}^5 $和$ \ MATHSF {U}^6 $ in $ \ MATHBB {f} _2^n $
Quantitative inverse theorem for Gowers uniformity norms $\mathsf{U}^5$ and $\mathsf{U}^6$ in $\mathbb{F}_2^n$
论文作者
论文摘要
我们证明了逆定理的定量范围，用于Gowers统一性规范$ \ MATHSF {U}^5 $和$ \ MATHSF {U}^6 $ in $ \ MATHBB {f} _2 _2^n $。证明从Gowers的早期部分结果和作者开始，将反问题减少到对某些多线性形式的代数性质的研究。本文中的大部分工作是对$ \ operatorname {sym} _4 $的自然动作之间的关系和$ \ operatatorName {sym} _5 $在多线性形式的空间与分区等级的空间，使用代数的常规性方法。一路上，我们对tidor的猜想给出了积极的答案，围绕5个变量中的近似对称的多线性形式，这在4个变量的情况下是错误的。最后，我们讨论了$ \ mathsf {u}^k $ norms的参数可能概括。
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 5 variables, which is known to be false in the case of 4 variables. Finally, we discuss the possible generalization of the argument for $\mathsf{U}^k$ norms.
