Minimal asymptotic \(g\)-adic bases from arbitrary partitions

Abstract: Let $g&gt;h\ge2$ be integers. For a nonempty set $W\subseteq\N$, let $\mathcal F<sup>*(W)$ denote the set of its finite nonempty subsets. Let $$A_g(W)=\left{\sum_{f\in F}a_fg^f: 1\leq a_f \leq g-1,F \in \mathcal F<sup>*(W)\right},$$ and let $\N=W_1\sqcup\cdots\sqcup W_h$ be a partition into infinite sets. We prove that $A=\bigcup_{i=1}^h A_g(W_i)$ is a minimal asymptotic basis of order $h$ for every such partition.