Approximate groups and their applications: work of Bourgain, Gamburd, Helfgott and Sarnak

Abstract

This is a survey of several exciting recent results in which techniques originating in the area known as additive combinatorics have been applied to give results in other areas, such as group theory, number theory and theoretical computer science. We begin with a discussion of the notion of an approximate group and also that of an approximate field, describing key results of Freiman-Ruzsa, Bourgain-Katz-Tao, Helfgott and others in which the structure of such objects is elucidated. We then move on to the applications. In particular we will look at the work of Bourgain and Gamburd on expansion properties of Cayley graphs on SL_2(F_p) and at its application in the work of Bourgain, Gamburd and Sarnak on nonlinear sieving problems.