A nilpotent Freiman dimension lemma

TL;DR

This paper extends Freiman's dimension lemma to torsion-free nilpotent groups, showing that approximate subgroups can be covered by a bounded number of cosets of a nilpotent subgroup with bounded rank, with explicit bounds.

Contribution

It introduces a nilpotent analogue of Freiman's dimension lemma, providing explicit bounds for covering approximate subgroups in torsion-free nilpotent groups.

Findings

Approximate subgroups are covered by bounded cosets of a nilpotent subgroup.

Explicit bounds depend only on the approximation parameter K.

The result generalizes Freiman's lemma to a nilpotent setting.

Abstract

We prove that a K-approximate subgroup of an arbitrary torsion-free nilpotent group can be covered by a bounded number of cosets of a nilpotent subgroup of bounded rank, where the bounds are explicit and depend only on K. The result can be seen as a nilpotent analogue to Freiman's dimension lemma.