Problems in additive and extremal combinatorics

这项研究将集中在一系列属于加性组合学的相关问题和概念上，加性组合学是介于解析数论、极值组合学、调和分析、动力系统和群论之间的一个数学领域。这些问题的共同特点是，它们往往是一种组合的味道，而且相当容易描述，解决这些问题的正确方法通常是从其他领域引入工具。在过去的二十多年里，这导致了一种非常富有成效的交叉受精，也产生了一些非常有趣的问题，本论文将尝试其中的一些问题。

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.