Abstract
In the fifth lecture, Ruzsa's folding lemma was proved. Next, Bohr sets were defined, and a connection was explained between Bohr sets and sets obtained by the intersection of a lattice with a symmetric convex set. This was followed by the statement and proof of Bogolyubov's lemma, which shows that, if A is a dense subset of a cyclic group, then the set A + A - A - A contains a large Bohr set of small dimension, which is sufficient to guarantee that it has a high degree of structure. Next, all these tools were brought together to prove a form of Freiman's theorem that is slightly weaker than the version usually stated, but sufficient for applications. This was followed by the statement of a theorem by Balog and Szemerédi, in a quantitative version discovered by me.
