Topics in harmonic analysis and additive combinatorics

该项目涉及调和分析和加法组合学两种不同类型的问题。第一次收集的问题将关注的理论的进一步发展的强奇异积分（线性算子的积分内核是过于奇异的起源是卡尔德龙-Zygmund型），特别是将集中使用振荡积分技术，以调查的规律性，这些经典运营商都沿着品种和不同的李群。第二个问题集将关注傅立叶分析技术在加法组合数学中的应用，特别是我们将关注与美丽的观察（由Sarkozy和Furstenberg独立完成）有关的定量结果，即具有正上密度的整数子集必然包含平方差。其中第一个是专门关注的泛化（和扩展到更一般的抽象设置）的一些著名的结果，从经典的谐波分析。第二个问题涉及证明密度拉姆齐理论结果的定量版本，该理论建立了大量整数点集中某些算术模式的存在性。

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.