The Freiman--Ruzsa Theorem over Finite Fields

TL;DR

This paper proves a conjecture by Ruzsa that bounds the size of subgroups containing sets with small doubling in finite abelian groups of prime torsion, extending previous results from the case r=2 to all primes.

Contribution

The paper confirms Ruzsa's conjecture for all prime torsion groups, providing a tight bound on subgroup size for sets with small doubling.

Findings

Confirmed Ruzsa's conjecture for prime torsion groups.

Established tight bounds on subgroup sizes.

Extended previous results from r=2 to all primes.

Abstract

Let G be a finite abelian group of torsion r and let A be a subset of G. The Freiman--Ruzsa theorem asserts that if |A+A| < K|A| then A is contained in a coset of a subgroup of G of size at most r^{K^4}K^2|A|. It was conjectured by Ruzsa that the subgroup size can be reduced to r^{CK}|A| for some absolute constant C >= 2. This conjecture was verified for r = 2 in a sequence of recent works, which have, in fact, yielded a tight bound. In this work, we establish the same conjecture for any prime torsion.