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

Source abstract

Let $h\geq2$ 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.