Harmonic analysis and arithmetic combinatorics

算术组合是一个新兴的数学领域，它结合了调和分析、解析和组合数论、组合学和遍历理论的问题、思想和方法。它是围绕一些中心问题组织起来的，比如在大集合中模式的存在性（例如等差数列），或者确定“极值”配置的结构（例如具有小集合的集合）。它最近的亮点包括高尔斯关于Szemer 'edi定理的工作和Green-Tao关于素数的长等差数列的结果。我提议的研究涉及调和分析与算术组合的各个方面之间的相互联系，特别是加性数论中的调和解析方法，调和分析与关联几何之间的联系，以及分析和几何测度理论中的算术组合方法。

Arithmetic combinatorics is an emerging area of mathematics which combines problems, ideas and methods from harmonic analysis, analytic and combinatorial number theory, combinatorics, and ergodic theory. It is organized around a number of central questions, such as the existence of patterns (e.g. arithmetic progressions) in large sets, or determining the structure of ``extremal" configurations (e.g. sets with small sumsets). Its recent highlights include the work of Gowers on Szemer\'edi's theorem and the Green-Tao result on long arithmetic progressions in the primes. My proposed research concerns the interconnections between harmonic analysis and various aspects of arithmetic combinatorics, in particular harmonic-analytic methods in additive number theory, connections between harmonic analysis and incidence geometry, and arithmetic-combinatorial methods in analysis and geometric measure theory.