On semilinear sets and asymptotic approximate groups

Let G be any group and A be a non-empty subset of G. The h-fold product set of A is defined asAh:={a1⋅a2⋯ah:a1,…,ah∈A}. Nathanson considered the concept of an asymptotic approximate group. Let r,l∈Z>0. 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|⩽l and Ar⊆XA. The set A is an asymptotic (r,l)-approximate group if the product set Ah 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). In this article, our motivations are three-fold:(1)We give an alternate proof of Nathanson's result.(2)From the alternate proof we deduce an improvement in the bound on the explicit constant l′.(3)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.