Small doublings in abelian groups of prime power torsion

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 $|A+A|\le K|A|$, 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.

We have decided to withdraw the paper, as its main result is subsumed by the authors and Kong in arXiv:2609.13021

References in corpus (1)

• Freiman-Ruzsa-type theory for small doubling constant