Some problems at the interface of harmonic analysis, number theory, and combinatorics

调和分析、数论和组合学接口的一些问题

• 批准号: 1600840
• 负责人: Akos Magyar
• 金额: $ 16.44万
• 依托单位: University of Georgia Research Foundation Inc
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600840&HistoricalAwards=false
• 关键词: Some; problems; interface; harmonic; analysis

So-called Ramsey theory deals with the problem of finding structures in large but otherwise disorganized sets. In the geometric setting it is to show that such sets contain a translated and rotated copy of a given finite set, or of its sufficiently large dilates. In other words it is to study the occurrence of geometric patterns. Over the past fifteen years there has been a remarkable progress of the study of linear patterns, developing and introducing tools from mathematical analysis, often referred to as higher-order Fourier analysis. Among the major achievements is the celebrated result of Green and Tao, which states that there are arbitrary long sequences of equally spaced prime numbers. This project builds on this development, and one of its major objectives is to develop analytic tools to understand the occurrence of geometric and arithmetic (i.e., defined by equations) structures in large but otherwise arbitrary sets. The problems arise in the context of the prime and integer lattice and also in classical Euclidean spaces. The principal investigator's approaches involve the interplay of techniques from discrete harmonic analysis and number theory, in addition to a new ingredient, ideas from additive combinatorics.The first motivational context for the project is that of prime numbers: to study nonlinear relations among the primes and to investigate the related problem of finding geometric constellations among points with prime coordinates. The underlying philosophy of considering the primes as a random subset of the integers leads naturally to the study of analogous questions in large sets of integer points and also in large measurable subsets of Euclidean spaces. Geometric structures in such sets are not well understood. The project aims to develop a general approach based on the modern point of view of additive combinatorics; namely, to establish appropriate notions of randomness that control the frequency at which a certain pattern occurs and to prove structure theorems for sets that are not suitably random. The underlying constructs are analytic and are related to objects studied in discrete harmonic analysis such as maximal operators and Radon transforms acting on functions defined on the integer lattice. Finally, the project aims to study geometric patterns in large measurable subsets of Euclidean spaces from this novel point of view, strengthening the connections between additive combinatorics and classical harmonic analysis.

所谓的拉姆齐理论解决的是在大型但杂乱的集合中寻找结构的问题。在几何设置中，它表明这些集合包含给定有限集合或其足够大的扩张的平移和旋转副本。换句话说就是研究几何图案的出现。在过去的十五年里，线性模式的研究取得了显着的进展，开发和引入了数学分析工具，通常称为高阶傅里叶分析。主要成就之一是格林和陶的著名结果，他们指出存在任意长的等距素数序列。该项目建立在这一发展的基础上，其主要目标之一是开发分析工具来理解大型但任意集合中几何和算术（即由方程定义）结构的出现。这些问题出现在素数和整数格的背景下以及经典欧几里得空间中。首席研究员的方法涉及离散调和分析和数论技术的相互作用，以及加性组合学思想这一新成分。该项目的第一个动机背景是素数：研究素数和素数之间的非线性关系。研究在具有素数坐标的点之间寻找几何星座的相关问题。将素数视为整数的随机子集的基本原理自然会导致对大型整数点集合以及欧几里得空间的大型可测量子集中的类似问题的研究。这些集合中的几何结构尚不清楚。 该项目旨在开发一种基于现代加性组合学观点的通用方法；即，建立适当的随机性概念来控制某种模式发生的频率，并证明不适当随机的集合的结构定理。底层构造是解析的，并且与离散调和分析中研究的对象相关，例如作用于整数晶格上定义的函数的最大算子和 Radon 变换。最后，该项目旨在从这个新颖的角度研究欧几里得空间的大型可测量子集中的几何图案，加强加法组合学和经典调和分析之间的联系。