Freiman's Theorem in an arbitrary abelian group

Abstract

A famous result of Freiman describes the structure of finite sets A of integers with small doubling property. If |A + A| <= K|A| then A is contained within a multidimensional arithmetic progression of dimension d(K) and size f(K)|A|. Here we prove an analogous statement valid for subsets of an arbitrary abelian group.

15 pages, to appear in London Math. Society journals. Exceptionally minor cosmetic changes from the previous version