The Circle Method as an Interface of Arithmetic Geometry, Additive Combinatorics and Harmonic Analysis

本提案的目的是说明和利用Hardy-Littlewood（圆）方法是算术调和分析的一种基本技术，它提供了一个连接算术几何、加性组合学和调和分析的强大接口。首先，在算术几何中，应用对角化技术研究了高维完全交点上有理点的密度，应用大筛不等式研究了完全交点代数族中的Hasse原理，并将圆法与下降法结合起来研究了各种中间维数上的有理点。在谐波分析的基础上，研究了多变量多项式丢芬图不等式解集的傅里叶变换，并将其应用于均匀分布和多项式遍历结果。第三，在与加性组合学的界面上，作者试图建立一种利用Gower的二次和更高均匀性思想的新版本的圆方法。一方面，这里的应用将影响算术几何的问题，另一方面，作者通过应用现代圆法理论中的技术，在高尔斯和格林-陶的工作中寻求定量改进。这个建议调查了数学的三个领域之间的接口：数论（更具体地说，丢番图问题），调和分析和加性组合。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，互联网和基于web的商业）。谐波分析研究了傅里叶分析的推广，傅里叶分析在更大的背景下在电气工程和通信中起着至关重要的作用。加性组合学试图理解相当一般的（因此看起来是非结构化的）集合中的底层结构，特别是当这些集合被算术运算修改时。本提案应用一种称为圆法的基本技术在这三个领域之间转移技术，既增强了我们在每个领域的知识，又增加了圆法作为算术谐波分析基本工具的范围。

The purpose of this proposal is to illustrate and exploit the observation that the Hardy-Littlewood (circle) method is a fundamental technique of arithmetic harmonic analysis that provides a powerful interface connecting arithmetic geometry, additive combinatorics and harmonic analysis. First, in arithmetic geometry, the proposer applies diagonalization techniques to investigate the density of rational points on complete intersections of high dimension, applies the large sieve inequality to study the Hasse principle in algebraic families of complete intersections, and combines the circle method with descent techniques so as to investigate rational points on varieties of intermediate dimension. On the interface with harmonic analysis, the proposer investigates the Fourier transform of the solution set of polynomial diophantine inequalities in many variables, with intended applications to uniform distribution and polynomial ergodic results. Third, on the interface with additive combinatorics, the proposer seeks to establish a novel version of the circle method that exploits Gower's quadratic and higher uniformity ideas. In one direction,applications here would impact questions in arithmetic geometry, and in another the proposer seeks quantitative improvements in the work of Gowers and Green-Tao by applying techniques from the modern theory of the circle method.This proposal investigates the interface between three areas of mathematics: number theory (and more specifically diophantine problems), harmonic analysis and additive combinatorics. 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 internet and web-based commerce). Harmonic analysis investigates generalizations of Fourier analysis, which in the larger setting plays a crucial role in electrical engineering and communications. Additive combinatorics seeks to understand the underlying structure in quite general (and seemingly, therefore, unstructured) sets, particularly as these sets are modified by arithmetic operations. This proposal applies a fundamental technique known as the circle method to transfer technology between these three areas, both enhancing our knowledge in each area and increasing the scope of the circle method as a basic tool of arithmetic harmonic analysis.