论文标题
戈尔斯统一性规范的定量逆理论
Quantitative inverse theory of Gowers uniformity norms
论文作者
论文摘要
（本文是为Bourbaki研讨会撰写的有关F. Manners工作的调查。） Gowers统一性规范是高阶傅立叶分析的中心对象，这是添加剂组合制剂的基石之一，并且在Gowers的Szemerédi的定理和绿色TAO定理中都起着重要作用。逆定理指出，如果一个函数具有较大的均匀性规范，这是结构的强大组合度量，则必须与nil序列相关，这是一个高度结构化的代数对象。格林，陶和齐格勒从定性的意义上证明了这一点，但证据是无能力提供合理界限的。在2018年，举止通过提供逆定理的新证明取得了突破。这个新的证明不仅包含大量的新见解，而且还首次提供了定量界限，而最糟糕的是双重指数。这次演讲将为反定理所说的话，为什么重要以及新的举止证明提供高级概述。
(This text is a survey written for the Bourbaki seminar on the work of F. Manners.) Gowers uniformity norms are the central objects of higher order Fourier analysis, one of the cornerstones of additive combinatorics, and play an important role in both Gowers' proof of Szemerédi's theorem and the Green-Tao theorem. The inverse theorem states that if a function has a large uniformity norm, which is a robust combinatorial measure of structure, then it must correlate with a nilsequence, which is a highly structured algebraic object. This was proved in a qualitative sense by Green, Tao, and Ziegler, but with a proof that was incapable of providing reasonable bounds. In 2018 Manners achieved a breakthrough by giving a new proof of the inverse theorem. Not only does this new proof contain a wealth of new insights but it also, for the first time, provides quantitative bounds, that are at worst only doubly exponential. This talk will give a high-level overview of what the inverse theorem says, why it is important, and the new proof of Manners.
