一、 課程背景與目的：

Additive Combinatorics is an active research topic which is in the interplay of Fourier analysis, combinatorics and number theory. Most of the important problems in this area are to explore the structures of a set A under some natural assumptions. One of the most known results is the existence of 3-term arithmetic progressions in a subset of integers with arbitrary positive upper density. This series of lectures is to give a detailed introduction for some selected topics such as sum-product phenomena, incidence geometry and their connections and applications to combinatorial problems.

 

二、 課程之大綱：

The lectures will mainly follow the book of Tao-Vu on additive combinatorics. The topics in the first part of the lectures contain sum set estimates, additive energy and Rusza distance formula followed by basic Fourier analysis on groups and its connections to additive combinatorics. The second part contains incidence geometry, including the powerful tool Szemeredi-Trotter points-lines incidences, sum-product estimates in integers and complex numbers as well as its applications to other combinatorial problems.