Problems in additive and extremal combinatorics

这项研究将集中在一组相关的问题和概念，属于添加剂组合，数学领域之间的接口解析数论，极值组合，调和分析，动力系统和群论。这些问题的共同特点是，它们往往具有组合的味道，而且相当容易陈述，解决这些问题的正确方法通常是从其他领域引入工具。在过去的20年左右，这导致了一个非常富有成效的交叉施肥，也带来了一些非常有趣的问题，其中一些将尝试在这篇论文。

This research will focus on a cluster of related problems and concepts that belong to additive combinatorics, an area of mathematics at the interface between analytic number theory, extremal combinatorics, harmonic analysis, dynamical systems, and group theory. The common feature of the problems is that they tend to be of a combinatorial flavour, and fairly easy to state, and that the right approach to these problems is usually to bring in tools from other areas. Over the last twenty years or so, this has led to a very fruitful cross-fertilization, and also to a number of very interesting questions, some of which will be attempted during this thesis.