On strongly minimal asymptotic bases of order $h$ generated by powers of $2$

Abstract

Let $h\ge 2$ be an integer and let ${\mathbb{N}}_0=W_1\cup \cdots \cup W_h$ be a partition into infinite sets. For $W\subseteq {\mathbb{N}}_0$, let $A(W)$ denote the set of all finite nonempty sums of distinct powers $2^w$ with $w\in W$. We give a necessary and sufficient condition for $A=A(W_1)\cup \cdots \cup A(W_h)$ to be a strongly minimal asymptotic basis of order $h$. This provides a complete answer to a problem posed by Nathanson in 1988.