Jean Bourgain
 

Abstract:   Sum-product theorems roughly express that starting from a given set A, either by considering the sumset A+A or the product set A.A, we obtain something significantly larger than A. This principle can be made precise in various contexts, some originating from attempts to progress on the higher dimensional Kakeya problem. The purely algebraic result in the context of a prime field has turned out to be an effective tool in establishing new bounds on exponential sums in situations where the usual Stepanov method seems ineffective. We will discuss new estimates on Gauss sums and sparse polynomials and their application to problems in cryptography.