On a problem of Nathanson related to minimal asymptotic bases and maximal asymptotic nonbases

Abstract

Let ${\mathbb{N}}_0=\{0,1,2,\dots \}$ and let $h\ge 2$ be an integer. For a set $A\subseteq {\mathbb{N}}_0$, write $hA$ for the set of all sums of $h$, not necessarily distinct, elements of $A$. In this paper, we prove that for every $h\ge 2$, there is a partition ${\mathbb{N}}_0=A\bigsqcup B$ such that $A$ is a minimal asymptotic basis of order $h$ and $B$ is a maximal asymptotic nonbasis of order $h$. This solves an open problem posed by Nathanson in 1974.