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

这一建议突出了哈代-利特尔伍德(圈)方法作为一种灵活的工具的作用，它支撑着与数论交叉的许多数学领域的进展。提出者打算继续进行一些调查，以说明该方法的跨学科方面，集中在算术几何和组合学的应用上。首先，研究人员构造了一大类代数变种的例子，这些例子的维度相对较小，可以用圆法来研究。关于有理点的密度和弱逼近的问题应该是可以理解的，其目的是阐明Manin和其他人的猜想。在第二个方向上，提出者应用现代丢番图近似理论的结果来改进高尔最近显式版本的Szmeredi定理。在一种更传统的询问方式中，研究人员调查整数表示为幂和的数量的分布。后者能够考虑比传统均方法更高的矩，导致以前无法达到的类型的多维应用。数字理论研究整数的性质。自古以来，丢番图方程(要以整数形式求解的方程)的研究形成了数论的核心组成部分，并在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)。矩问题的研究是傅立叶分析的一个基本问题，在更大的背景下，它在电气工程和通信中起着至关重要的作用。圆周方法的现代化身提供了一种将技术从一个领域转移到另一个领域的机制，这项提议致力于促进这种转移的基础研究。

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.