Additive combinatorics

October - November 2025

Synopsis

Additive combinatorics is the study of additive questions about finite sets of integers. Possible topics include the theorems of Roth and Szemerédi (on additive progressions in dense sets of integers), Freiman's theorem and its relatives (on the structure of sumsets that are almost closed under addition), sum-product phenomena, applications to exponential sum estimates of Bourgain et al over multiplicative subgroups of finite fields, and to the Littlewood-Offord problem. A particular aim of the course will be to give a taster of the varied methods which have been brought to bear on the subject: Fourier analysis, polynomial methods, probabilistic methods, methods from information theory, graph theory and incidence geometry. This course will have two parts:

• Part I (Browning) will focus more on the number theory
• Part II (Kwan) will focus more on the combinatorics

Timetable

• Part I: Oct 7, Oct 9, Oct 21, Oct 23
• Part II: Oct 28, Oct 30, Nov 4, Nov 6, Nov 11, Nov 13

Course Assessment

The course will be assessed by problem sheets.