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

这个建议强调了哈代-利特尔伍德（圆）方法作为一种灵活工具的作用，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调查，说明这一跨学科方面的方法，集中在算术几何和组合数学的应用。首先，研究人员构造了一个大类的例子代数簇，相对较小的维度方面，他们的程度，可以通过研究的圆的方法。关于有理点的密度和弱逼近的问题应该是容易理解的，目的是为了阐明Manin等人的观点。在第二个方向，提议者适用于结果从现代理论的丢番图近似，以完善高尔最近明确版本的塞梅雷迪定理。在一个更传统的调查路线中，研究者调查了整数表示的数量的分布作为幂和。后者的工作使更高的时刻比传统的均方被认为是，导致以前无法访问类型的多维应用。数论研究整数（"整数“）的属性。自古以来，丢番图方程的研究已经成为数论的一个核心组成部分，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）。矩问题的研究是傅立叶分析的一个基本课题，在更大的背景下，傅立叶分析在电气工程和通信中起着至关重要的作用。 循环方法的现代体现为将技术从这些领域之一转移到另一个领域提供了一种机制，而本建议则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.