Characterization of binary digit partitions yielding minimal asymptotic bases

Characterize all partitions \(\mathbb{N}=W_1\sqcup\cdots\sqcup W_h\) into infinite sets for which \(\bigcup_{i=1}^h A_2(W_i)\) is a minimal asymptotic basis of order \(h\).

Background

Nathanson’s construction establishes that the partition of $\mathbb{N}$ into residue classes modulo $h$ produces a thin, strongly minimal asymptotic basis of order $h$ through the binary sets $A_2(W_i)$. The paper describes subsequent progress, including a complete characterization for $h=2$, but indicates that a general characterization remains unresolved. The problem concerns identifying the precise partition-theoretic conditions that determine minimality for binary digit-partition constructions.

The related question of whether minimality holds for almost all partitions is posed in Nathanson’s Problem 1, but the explicit uncertainty marker in the cited passage directly supports the unresolved status of the general characterization.