Topics in harmonic analysis and additive combinatorics

调和分析和加法组合学主题

• 批准号: 0707099
• 负责人: Neil Lyall
• 金额: $ 7.1万
• 依托单位: University of Georgia Research Foundation Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-06-01 至 2011-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0707099&HistoricalAwards=false
• 关键词: Topics; harmonic; analysis; additive; combinatorics

The project involves work on two distinct types of problem in harmonic analysis and additive combinatorics. The first collection of problems will be concerned with on the further development of the theory of strongly singular integrals (linear operators whose integral kernels are too singular at the origin to be of Calderon-Zygmund type) and in particular will focus the use of oscillatory integral techniques to investigate the regularity of these classical operators both along varieties and on different Lie groups. The second collection of problems will concern the application of Fourier analytic techniques to problems in additive combinatorics, in particular we will focus on quantitative results relating to the beautiful observation (made independently by Sarkozy and Furstenberg) that subsets of the integers with positive upper density necessarily contain square differences.Broadly speaking the project will focus on two main problems. The first of these is specifically concerned with the generalization (and extension to more general abstract settings) of some well known results from classical harmonic analysis. The second problem relates to proving quantitative versions of results from density Ramsey theory establishing the existence of certain arithmetic patterns in large sets of integer points.

该项目涉及调和分析和加法组合学中两种不同类型的问题。第一组问题将涉及强奇异积分理论的进一步发展（线性算子，其积分核在原点上太奇异，不属于 Calderon-Zygmund 类型），特别将重点使用振荡积分技术来研究这些经典算子沿簇和不同李群的正则性。第二组问题将涉及傅立叶分析技术在加性组合问题中的应用，特别是我们将重点关注与美丽观察（由萨科齐和弗斯滕伯格独立完成）相关的定量结果，即具有正上密度的整数子集必然包含平方差。从广义上讲，该项目将关注两个主要问题。其中第一个具体涉及经典调和分析的一些众所周知的结果的概括（以及扩展到更一般的抽象设置）。第二个问题涉及证明密度拉姆齐理论结果的定量版本，确定大型整数点集中某些算术模式的存在。