A note on a result of Nathanson

Let $h\geq2$ be a positive integer. Let $W$ be a nonempty subset of $\mathbb{N}$. Denote by $\mathcal{F}^{*}(W)$ the set of all finite, nonempty subsets of $W$. Let $A(W)$ be the set of all numbers of the form $\sum\limits_{f\in F}2^f$, where $F\in \mathcal{F}^{*}(W)$. Is the asymptotic basis $A=\cup_{i=1}^h A(W_i)$ minimal for any partition $\mathbb{N}=W_1\cup\dots \cup W_h$? Nathanson [Minimal bases and powers of $2$, <span class="italic">Acta Arith.</span> <span class="bold">49</span> (1988), 525—532] showed that this is false for $h=2$. In this paper, we consider this problem for all $h\geq 2$.