Structure theory for measure-preserving systems, additive combinatorics, and correlations of multiplicative functions

考虑一个数字数据流——一个0和1的序列。这个序列可以是高度结构化的——例如，它可以在0和1之间周期性地交替。或者它可能是完全随机的,每个成员的价值序列的没有任何关系。它也可能是“伪随机”——由确定性算法描述，但在统计上与真正的随机序列无法区分。也可以是一些复杂的混合结构和伪随机性。我们能否精确地定义结构和随机性的含义，并将任意数据描述为这两种不同成分的组合？这些问题的重要性在密码学、计算机科学、组合学、动力学、和数论,因为它们允许一个数学确定某些模式任意数据流,保证发生与否。例如，2004年，本·格林（Ben Green）和PI解决了数论中一个长期存在的猜想，即质数包含任意长的等差数列，其关键思想是将质数分解为结构化和随机的组成部分，并研究每个组成部分的贡献。例如，在计算机科学中，这一理论已经导致了为几种类型的应用生成伪随机比特的有效方法。在随后的二十年里，在更精确地量化结构和随机性的含义方面取得了很大进展，特别是在现在被称为高阶傅立叶分析的数学领域。人们对数论结构（如质数）在大尺度和小尺度上表现出（伪）随机行为的精确方式有了更多的了解。有近年来在这个方向上取得稳定的进展,在规模上是哪一个能够明确说明各种类型的伪随机数已经随着时间的推移,缩小和进一步的工作将在这个项目中,特别是,它如此地接近解决(版本)的一个著名的猜想在数论Chowla猜想——这可以反过来垫脚石更著名猜想如双胞胎'猜想。本项目为研究生提供研究训练机会。在这个项目中，PI（与合作者一起）计划开展两个相关项目。首先，PI将继续最近的工作，一方面发展加性组合中的Gowers均匀性范数的一般逆定理，另一方面发展遍历理论中的Host- Kra均匀性半范数。其次，PI将继续建立在最近对乘法函数的理解上的突破，在对这些函数的（对数平均）Chowla和Elliott猜想方面取得进一步进展，并将这些结果应用于解析数论中的相关问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Consider a stream of digital data - a sequence of zeroes and ones. This sequence could be highly structured - for instance, it could alternate periodically between 0 and 1. Or it could be completely random, with the value of each member of the sequence having no relation whatsoever to the next. It could also be "pseudorandom" - described by a deterministic algorithm, but yet statistically indistinguishable from a genuinely random sequence. Or it could be some complex mixture of structure and (pseudo)randomness. Can one define precisely what structure and randomness mean and describe arbitrary data as combinations of these two different components? Such questions are of importance in cryptography, computer science, combinatorics, dynamics, and number theory, as they allow one to mathematically determine whether certain patterns in arbitrary streams of data are guaranteed to occur or not. For instance, in 2004, Ben Green and the PI were able to settle a long-standing conjecture in number theory that the prime numbers contained arbitrarily long arithmetic progressions, with the key idea being to break up the prime numbers into structured and random components and study the contribution of each component. In computer science, this theory has led, for instance, to efficient ways to generate pseudorandom bits for several types of applications. In the subsequent twenty years, much progress has been made in quantifying more precisely what structure and randomness mean, particularly in the area of mathematics now known as higher-order Fourier analysis. More understanding has been gained on the precise way in which number-theoretic structures, such as the primes, exhibit (pseudo-)random behavior at both large and small scales. There has been steady progress in this direction in recent years, in which the scale on which one is able to definitively demonstrate various types of pseudorandomness has narrowed over time, and further work will be carried out in this project, in particular, it is tantalizingly near to resolve (a version) of a well-known conjecture in number theory - the Chowla conjecture - which could be in turn a stepping stone to even more famous conjectures such as the twin prime conjecture. This project provides research training opportunities for graduate students. In this project, the PI (in conjunction with collaborators) plans to work on two related projects. Firstly, the PI will continue recent work on developing general inverse theorems for the Gowers uniformity norms in additive combinatorics on one hand and the Host--Kra uniformity seminorms in ergodic theory on the other. Secondly, the PI will continue building upon recent breakthroughs in the understanding of multiplicative functions, to make further progress towards the (logarithmically averaged) Chowla and Elliott conjectures for such functions, and to apply these results to related problems in analytic number theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.