Abstract
The fourth lecture was an introduction to the theory of sums of sets of integers. The first result presented was a theorem of Khovanskii, which shows that, for any finite set A, the sizes of the iterated sum sets A + A + ... + A depend on the number of copies of the set A polynomially once k is sufficiently large. Curiously, the proof gives no information on how large k must be before polynomial behavior begins, but more recent results have provided explicit bounds. Freiman's theorem was then stated and some of the steps in the proof were presented. These included basic facts about Freiman homomorphisms and the statement of Ruzsa's plunge lemma, which allows a large part of a set whose sum is small to be plunged into a cyclic group not much larger.
