Description

Additive combinatorics is a relatively modern and vibrant area of mathematics that tries to understand the additive structures in subsets of integers. A classic example is provided by a result by Roth (1954) which states that every subset of integers of positive upper density contains a three-term arithmetic progression. This exciting area combines techniques from analysis, number theory as well as combinatorics. The aim of the project is to provide an introduction to this area, including detailed treatments of the following topics connected to questions 1)-3):

a) Roth's theorem on 3-term arithmetic progressions

b) sumsets and the Freiman-Ruzsa theorem

c) the sum-product phenomenon

Our approach will be heavily based on examples, which is anticipated to provide avenues for independent exploration. The flavour of the techniques used will be analytic in nature.

Prerequisite

Resources

T. Tao and V. Vu., Additive Combinatorics.

Davenport H. Analytic Methods for Diophantine Equations and Diophantine Inequalities, (Chapter 3), Cambridge university press, ISBN: 9780521605830.

S. Prendiville. Roth's theorem- an exposition (2008).