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

Abstract

Let $\mathbb{N}_0=\{0,1,2,\ldots\}$ and let $h\ge2$ 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 $h\ge2$, there is a partition $\mathbb{N}_0=A\sqcup B$ 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.