Problems in additive combinatorics

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My main field of research is in Additive Combinatorics. The central problem of the area is to characterize additive or multiplicative structures. I am especially interested in the "incompatibility" of multiplicative and additive structures. An old conjecture of Erdos and Szemeredi states that if A is a finite set of integers then the sum-set or the product-set should be large. The sum-set of A is defined as A+A={a+b | a,b \in A} and the product set is A*A={ab | a,b \in A}. Erdos and Szemeredi conjectured that the sum-set or the product set is almost quadratic in the size of A, i.e. |A+A|+|A*A|~|A||A|. The best known lower bound on the quantity |A+A|+|A*A| follows from an earlier work of mine, however it is still very far from the conjecture. The same problem was considered by Bourgain, Katz, and Tao, for the finite field case. Their result has important applications , not only to number theory, but to computer science, Ramsey theory, and cryptography. It turned out that methods from discrete geometry can be used efficiently not only for the real/complex case but for the finite field case as well.