Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On semilinear sets and asymptotically approximate groups   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.

4
The categories of this paper are: cond-mat
12
The categories of this paper are: nucl-th
21
The categories of this paper are: adap-org
19
The categories of this paper are: eess
1
The categories of this paper are: cs
3
The categories of this paper are: physics
35
The categories of this paper are: mtrl-th
26
The categories of this paper are: bayes-an
29
The categories of this paper are: funct-an
20
The categories of this paper are: acc-phys
24
The categories of this paper are: dg-ga
11
The categories of this paper are: hep-lat
27
The categories of this paper are: comp-gas
31
The categories of this paper are: ao-sci
33
The categories of this paper are: plasm-ph
15
The categories of this paper are: nucl-ex