Oriol Serra.

Titre: Doubling and Volume: A conjecture of Freiman

The celebrated Theorem of Freiman states that a set A of integers with  
small doubling, |A+A|<c|A|,   is a dense subset of a multidimensional  
arithmetic progression. Lower bounds for the density of A in the  
progression in terms of c have been improved by several authors,  
including Freiman, Bilu, Chang, Sanders, Konyagin and Schoen.

An approach to the problem of determining the optimal value of this  
lower bound is to determine the maximum volume of a set  with given  
cardinality and doubling. The volume of a set A of integers is the  
smallest volume of the convex hull, this minimum  taken among all sets  
which are Freiman isomorphic to A.

The so--called (3k-4)-Theorem gives a precise answer to the problem of  
determinig the maximum volume of a set A with given cardinality and  
doubling |A+A|<c|A| when c<3. Freiman recently conjectured a formula  
for this maximum volume for a general value  of the constant c.

The talk will discuss recent advancements in this direction, in  
particular proving the conjecture for a class of sets of integers  
called chains, which contain a nested sequence of extremal sets of the  
same nature. Some extensions and implications will be also discussed.

This is joint work with G.A. Freiman.