Applications of the Hardy-Littlewood Method in Number Theory and Beyond

Hardy-Littlewood 方法在数论及其他领域的应用

• 批准号: 0140523
• 负责人: Trevor Wooley
• 金额: --
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-01 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0140523&HistoricalAwards=false
• 关键词: Applications; Hardy; Littlewood; Method; Number

This proposal highlights the role of the Hardy-Littlewood (circle) method as a flexible tool that underpins progress in many areas of mathematics intersecting with number theory. The proposer intends to pursue a number of investigations that illustrate this intersubdisciplinary aspect of the method, concentrating on applications in arithmetic geometry and combinatorics. First, the researcher constructs a large class of examples of algebraic varieties, of relatively small dimension in terms of their degree, that may be studied by means of the circle method. Questions concerning the density of rational points and weak approximation should be accessible,the intention being to shed light on conjectures of Manin andothers. In a second direction, the proposer applies results from the modern theory of diophantine approximation in order to refine Gower's recent explicit version of Szemeredi's theorem. In a more traditional line of enquiry, the researcherinvestigates the distribution of the number of representations of integers as sums of powers. The latter work enables higher moments than the traditional mean square to be considered, leading to multidimensional applications of previously inaccessible type.Number Theory studies the properties of integers (``whole numbers''). Since antiquity, the study of diophantine equations (equations to be solved in integers) has formed a core component of Number Theory, and has recently influenced the development of codes and cryptosystems (applied, for example, in data storage systems such as compact disks and DVDs, communications systems and banking security). The investigation of moment problems is a fundamental topic for Fourier analysis, which in the larger setting plays a crucial role in electrical engineering and communications. The circle method, in its modern incarnations, provides a mechanism for transferring technology from one of these areas to another, and this proposal pursues basic research that facilitates such transfers.

这一建议强调了Hardy-Littlewood（圆）方法作为一种灵活的工具的作用，它支撑了与数论交叉的许多数学领域的进步。提议者打算进行一些研究，以说明该方法的跨学科方面，集中在算术几何和组合学中的应用。首先，研究者构造了一大类代数变量的例子，它们的度数相对较小，可以用圆法来研究。关于有理点的密度和弱近似的问题应该是可访问的，其目的是阐明马宁和其他人的猜想。在第二个方向上，提议者应用丢芬图近似值的现代理论的结果，以完善高尔最近对塞梅雷迪定理的明确版本。在更传统的研究路线中，研究人员调查整数作为幂和的表示的数量分布。后一种工作可以考虑比传统均方更高的矩，从而实现以前无法实现的多维应用类型。数论研究整数的性质（“整数”）。自古以来，对丢芬图方程（以整数形式求解的方程）的研究已经形成了数论的核心组成部分，并且最近影响了代码和密码系统的发展（例如，应用于数据存储系统，如光盘和dvd，通信系统和银行安全）。矩问题的研究是傅里叶分析的一个基本主题，在更大的背景下，傅里叶分析在电气工程和通信中起着至关重要的作用。现代的循环方法提供了一种将技术从一个领域转移到另一个领域的机制，而本提案追求的是促进这种转移的基础研究。