On a problem of Nathanson related to minimal asymptotic bases of order $h$

Abstract

For integer $h\ge 2$ and $A\subseteq \mathbb{N}$, we define $hA$ to be all integers which can be written as a sum of $h$ elements of $A$. The set $A$ is called an asymptotic basis of order $h$ if $n\in hA$ for all sufficiently large integers $n$. An asymptotic basis $A$ of order $h$ is minimal if no proper subset of $A$ is an asymptotic basis of order $h$. For $W\subseteq \mathbb{N}$, denote by ${\mathcal{F}}^{\ast }(W)$ the set of all finite, nonempty subsets of $W$. Let $A(W)$ be the set of all numbers of the form ${\sum }_{f\in F}{2}^f$, where $F\in {\mathcal{F}}^{\ast }(W)$. In this paper, we give some characterizations of the partitions $\mathbb{N}=W_1\cup \cdots \cup W_h$ with the property that $A=A(W_1)\cup \cdots \cup A(W_h)$ is a minimal asymptotic basis of order $h$. This generalizes a result of Chen and Chen, recent result of Ling and Tang, and also recent result of Sun.

8 pages