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

TL;DR

This survey reviews recent advances in additive combinatorics, focusing on approximate groups and fields, and their applications to group expansion, number theory, and computer science, highlighting key results by Bourgain, Gamburd, Helfgott, and Sarnak.

Contribution

It synthesizes recent developments in the structure of approximate groups and fields and their diverse applications across mathematics and computer science.

Findings

Structure theorems for approximate groups and fields

Expansion properties of Cayley graphs on SL_2(F_p)

Applications to nonlinear sieving problems

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.