On asymptotic approximate groups in nilpotent groups
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909033074524160 |
|---|---|
| author | Biswas, Arindam |
| author_facet | Biswas, Arindam |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_10691 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On asymptotic approximate groups in nilpotent groups Biswas, Arindam Group Theory 20F69, 20F18, 11B13, 11P70 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. |
| title | On asymptotic approximate groups in nilpotent groups |
| topic | Group Theory 20F69, 20F18, 11B13, 11P70 |
| url | https://arxiv.org/abs/2605.10691 |