Introduction to additive combinatorics lecture 10.1 --- the structure and properties of Bohr sets.

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An important informal idea in additive combinatorics is that of a "structured" set. One example of a class of sets that are rich in additive structure is the class of Bohr sets, which play the role in general finite Abelian groups that subspaces play in the special case of groups of the form F_p^n for fixed p and large n. Here I prove a few basic lemmas about them.

0:00 Introduction
1:24 A lower bound on the size of Bohr sets
8:20 Adding two Bohr sets
10:35 Bohr sets contain long arithmetic progressions
18:56 A Bohr set is isomorphic to the intersection of a cube with a lattice

Timothy Gowers

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