Additive Combinatorics: Sets, Finite Sequences and Remarkable Applications – CAESAR

The field of additive combinatorics is motivated by questions concerning additive structure in sets of integers, and more recently in subsets of more general groups. It is characterized by the wide range of mathematical tools that are employed to solve these problems, ranging from the algebraic through the combinatorial and probabilistic to the analytic.

Historically, Lagrange's four squares theorem was one of the first results in this direction, followed by important work of Cauchy, Goldbach, Waring and Davenport, amongst others. More recent highlights include Freiman's work on the structure of sets with small sumsets, Szemeredi's theorem on long arithmetic progressions in dense sets; the Green-Tao theorem on long arithmetic progressions in the primes; and a proof of Kemnitz's conjecture, all of which have attracted much international attention.

The research to be conducted within this project concerns various problems involving sets and sequences in groups -- mostly in abelian groups, though non-commutative problems also appear. As far as ``sets'' are concerned, we will seek improved lower bounds for the set of subset sums; we will explore expanders and problems related with Freiman models; and we will use analytic approaches to investigate questions related to sieves, sets lacking arithmetic structure, higher-degree uniformity and additive bases. Regarding ``sequences'', we will focus on zero-sum problems over finite abelian groups, including several problems related to the Erdos-Ginzburg-Ziv Theorem; the value of the Davenport constant for certain groups and their associated inverse problems; and several direct and inverse problems for the cross number, with applications to zero-sum problems that arise in non-unique factorization theory.

A regular seminar, one (or two) international conference(s) and a website -- to be organized and maintained as part of the project -- will serve as a forum for the exchange of ideas and the dissemination of results obtained on these and closely related topics.