On asymptotic approximate groups in nilpotent groups

TL;DR

This paper investigates the properties of asymptotic approximate groups within virtually nilpotent groups, establishing conditions under which finite sets exhibit approximate group behavior and extending results to certain infinite sets.

Contribution

It demonstrates that finite sets containing large symmetric word balls are asymptotic approximate groups in virtually nilpotent groups and extends to nonabelian semilinear sets.

Findings

Finite sets with large symmetric word balls are asymptotic approximate groups.

Established a nonabelian semilinear-set analogue for infinite sets.

Results apply to virtually nilpotent groups, broadening understanding of approximate groups.

Abstract

Let $G$ be a group and let $A\subseteq G$ be non-empty. We call $A$ an asymptotic $(r,l)$-approximate group if, for a fixed dilation factor $r$, the larger product sets $A^{hr}$ can, for all sufficiently large $h$, be covered by a bounded number of left translates of $A^h$, with the bound $l$ independent of $h$. We show that, in virtually nilpotent groups, finite sets whose powers contain a symmetric word ball of radius comparable to $h$ are asymptotic approximate groups. We also prove a nonabelian semilinear-set analogue for certain infinite sets in these groups.