Harmonic analysis and arithmetic combinatorics

算术组合学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、组合学和遍历理论中的问题、思想和方法。它围绕一些核心问题组织起来，例如在大集合中模式的存在(例如算术级数)，或确定“极值”构型(例如具有小和集的集合)的结构。它最近的亮点包括Gowers关于Szemer\‘EDI定理的工作和Green-Tao关于素数中长算术级数的结果。我建议的研究涉及调和分析与算术组合学的各个方面之间的相互联系，特别是加数理论中的调和分析方法，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.