# Small doublings in abelian groups of prime power torsion

## TL;DR

This paper proves a conjecture by Ruzsa regarding the structure of subsets with small doubling in finite abelian groups of prime power torsion, extending previous results to this broader class.

## Contribution

It confirms Ruzsa's conjecture for groups with prime power torsion, providing a tight bound and extending prior work from prime torsion cases.

## Key findings

- Confirmed Ruzsa's conjecture for prime power torsion groups.
- Established tight bounds for subset containment in cosets.
- Extended previous results from prime to prime power torsion groups.

## Abstract

Let $A$ be a subset of $G$, where $G$ is a finite abelian group of torsion $r$. It was conjectured by Ruzsa that if $|A+A|\leq K|A|$, then $A$ is contained in a coset of $G$ of size at most $r^{CK}|A|$ for some constant $C$. The case $r=2$ received considerable attention in a sequence of papers, and was resolved by Green and Tao. Recently, Even-Zohar and Lovett settled the case when $r$ is a prime. In this paper, we confirm the conjecture when $r$ is a power of prime. In particular, the bound we obtain is tight.