Small doubling in groups with moderate torsion

TL;DR

This paper characterizes the structure of finite subsets in abelian groups with small doubling, showing they are contained in either small coset progressions or unions of few cosets, extending Freiman's theorem to groups with moderate torsion.

Contribution

It extends Freiman's $(3n-3)$-theorem to arbitrary abelian groups with moderate torsion, providing sharp bounds and structural descriptions for sets with small doubling.

Findings

Sets with small doubling are contained in small coset progressions or few cosets of a finite subgroup.

Bounds on doubling and number of cosets are proven to be optimal and cannot be relaxed.

The result generalizes Freiman's theorem to a broader class of abelian groups.

Abstract

We determine the structure of a finite subset $A$ of an abelian group given that $|2A|<3(1-\epsilon)|A|$, $\epsilon>0$; namely, we show that $A$ is contained either in a "small" one-dimensional coset progression, or in a union of fewer than $\epsilon^{-1}$ cosets of a finite subgroup. The bounds $3(1-\epsilon)|A|$ and $\epsilon^{-1}$ are best possible in the sense that none of them can be relaxed without tightened another one, and the estimate obtained for the size of the coset progression containing $A$ is sharp. In the case where the underlying group is infinite cyclic, our result reduces to the well-known Freiman's $(3n-3)$-theorem; the former thus can be considered as an extension of the latter onto arbitrary abelian groups, provided that there is "not too much torsion involved".