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

保测系统的结构理论、加法组合学和乘法函数的相关性

• 批准号: 2347850
• 负责人: Terence Tao
• 金额: $ 75.34万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2347850&HistoricalAwards=false
• 关键词: Structure; theory; measure; preserving; systems; Structure; theory; measure; preserving; systems

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.

考虑一系列数字数据流 - 一个零和一个序列。 该序列可以高度结构化 - 例如，它可以定期在0到1之间进行交替。或者可以是完全随机的，而序列的每个成员的值与下一个没有任何关系。 它也可以是由确定性算法描述的“伪界”，但在统计学上与真正的随机序列无法区分。 或者它可能是结构和（伪）随机性的复杂混合物。 一个人可以准确定义什么结构和随机性含义并将任意数据描述为这两个不同组成部分的组合？ 这些问题在密码学，计算机科学，组合学，动态和数字理论中至关重要，因为它们可以在数学上确定是否保证了数据中的某些模式是否会发生。 例如，在2004年，本·格林（Ben Green）和PI能够解决一个长期以来的猜想，即质子数包含任意长的算术渐进性，关键思想是将质量数分解为结构化和随机组成部分，并研究每个组件的贡献。 在计算机科学中，该理论导致了例如有效的方法来为几种类型的应用生成伪随机位。在随后的二十年中，在更精确地量化结构和随机性方面取得了很多进展，尤其是在现在称为高阶傅立叶分析的数学领域。 在大小和小尺度上表现出（伪 - ）随机行为等数量理论结构（例如素数）的精确方式。近年来，在这个方向上取得了稳步的进步，在这种方向上，人们能够确定证明各种类型的伪随时性的规模随着时间的流逝而缩小，并且在这个项目中将进行进一步的工作，尤其是在这个项目中，它诱人地解决了（一种版本）（一种版本）在数字理论中是一个众所周知的构想 - 奇特拉（Chowla）的构想 - 这是一个众所周知的构想 - 这是一个众所周知的猜想，这是一个奇特的猜想，这是一个奇特的构想，这是一个偶然的猜想。猜想。该项目为研究生提供了研究培训机会。在这个项目中，PI（与合作者一起）计划在两个相关项目上工作。 首先，PI将继续进行最新的工作，以一方面为Gowers均匀性规范开发一般的倒数定理，另一方面是宿主-KRA均匀性半神经中的eRgodic理论。 其次，PI将继续基于最近对乘法功能的突破，进一步取得进一步的进步，以实现（对数平均）的Chowla和Elliott猜想，并将这些结果应用于分析数理论的相关问题。这一奖项在NSF的法定任务中反映了通过评估的范围，这一奖项反映了构成的范围。