Small doubling in groups

TL;DR

This paper surveys the theory of small doubling sets in groups, focusing on subsets where the product set is at most a constant multiple of the original set, including the well-known case of integers.

Contribution

It provides an overview of the theory of small doubling in groups, highlighting key results like the Freiman--Ruzsa theorem for integers.

Findings

Characterization of small doubling sets in various groups

Connections to additive combinatorics and structure theory

Overview of key theorems and open problems

Abstract

Let A be a subset of a group G = (G,.). We will survey the theory of sets A with the property that |A.A| <= K|A|, where A.A = {a_1 a_2 : a_1, a_2 in A}. The case G = (Z,+) is the famous Freiman--Ruzsa theorem.