The positive integers do not lie, like the logical foundations of mathematics, in the scarcely visible distance, nor in the uncomfortably tangled foreground, like the immediate data of the physical world, but a decent middle distance, where the outlines are clear and yet some element of mystery remains...There is no one so blind that he does not see them, and no one so sharp-sighted that his vision does not fail; they stand there a continual and inevitable challenge to the curiosity of every healthy mind. I have merely directed your attention for a moment to a few of the less immediately conspicuous features of the landscape, in the hope that I may sharpen your curiosity a little, and that some may feel tempted to walk a little nearer and take a closer view.

G. H. Hardy

This is the course page for the Part III course Additive Combinatorics, Lent term 2021. Lecture Notes: A PDF version of the lecture notes will be available on this website and the Moodle page. This will be updated periodically throughout the course, and attempts to be a complete record of everything lectured.

The current lectures notes are available here.

Problem sheets:

Other sources

Books: Additive combinatorics is a relatively young field, and there are not many books available. Parts of this course have only appeared in research papers, but complete lecture notes will be provided. Some excellent books which cover some (but not all) of the topics in this course are Geroldinger and Ruzsa's Combinatorial Number Theory and Additive Group Theory, Nathanson's Additive Number Theory: Inverse Problems and the Geometry of Sumsets, and Tao and Vu's Additive Combinatorics.

Alternative lecture notes: For the enthusiast, here is a collection of various notes on additive combinatorics that are freely available. No guarantee is made about the intersection of the topics covered in this course with what is presented in the following.