On semilinear sets and asymptotically approximate groups

TL;DR

This paper explores the structure of asymptotic approximate groups within arbitrary groups, extending previous results by showing unions of generalized arithmetic progressions form such groups.

Contribution

It generalizes Nathanson's result by demonstrating that unions of multiple generalized arithmetic progressions are asymptotic approximate groups in any abelian group.

Findings

Unions of k generalized arithmetic progressions are asymptotic (r, (4rk)^k)-approximate groups.

Every finite subset in an abelian group is an asymptotic approximate group.

The result applies to arbitrary abelian groups, not just specific cases.

Abstract

Let $G$ be any group and $A$ be an arbitrary subset of $G$ (not necessarily symmetric and not necessarily containing the identity). The $h$-fold product set of $A$ is defined as $$A^{h} :=\lbrace a_{1}.a_{2}...a_{h} : a_{1},\ldots,a_n \in A \rbrace.$$ Nathanson considered the concept of an asymptotic approximate group. Let $r,l \in \mathbb{N}$. The set $A$ is said to be an $(r,l)$ approximate group in $G$ if there exists a subset $X$ in $G$ such that $|X|\leqslant l$ and $A^{r}\subseteq XA$. The set $A$ is an asymptotic $(r,l)$-approximate group if the product set $A^{h}$ is an $(r,l)$-approximate group for all sufficiently large $h$. Recently, Nathanson showed that every finite subset $A$ of an abelian group is an asymptotic $(r,l')$ approximate group (with the constant $l'$ explicitly depending on $r$ and $A$). We generalise the result and show that, in an arbitrary abelian group $G$, the union of $k$ (unbounded) generalised arithmetic progressions is an asymptotic $(r,(4rk)^k)$-approximate group.