Freiman's theorem in an arbitrary nilpotent group

TL;DR

This paper extends Freiman's theorem to arbitrary nilpotent groups, showing that approximate subgroups are contained in structured coset nilprogressions with bounds depending only on the approximation parameter.

Contribution

It provides a new proof of a Freiman-Ruzsa-type theorem in nilpotent groups, avoiding Mal'cev's embedding, and establishes bounds depending solely on the step of the group.

Findings

Approximate subgroups are contained in coset nilprogressions with polynomial bounds.

The bounds depend only on the approximation parameter and the group's step.

A direct proof for torsion-free nilpotent groups is provided.

Abstract

We prove a Freiman-Ruzsa-type theorem valid in an arbitrary nilpotent group. Specifically, we show that a K-approximate subgroup A of an s-step nilpotent group G is contained in a coset nilprogression of rank at most f(K) and cardinality at most exp(g(K))|A|, with f and g polynomials depending only on the step s of G. To motivate this, we give a direct proof of Breuillard and Green's analogous result for torsion-free nilpotent groups, avoiding the use of Mal'cev's embedding theorem.