On semilinear sets and asymptotically approximate groups

Arindam Biswas, Wolfgang Alexander Moens

Published 2019-02-15Version 1

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.