On minimal asymptotic bases

Let N denote the set of all nonnegative integers. Let W be a nonempty subset of N. Denote by F∗(W) the set of all finite, nonempty subsets of W. Let A(W) be the set of all numbers of the form ∑f∈F2f, where F∈F∗(W). Let N=W1∪W2 be a partition with 0∈W1 such that W1 and W2 are infinite. In this paper, we prove that A=A(W1)∪A(W2) is a minimal asymptotic basis of order 2 if and only if either W1 contains no consecutive integers or W2 contains consecutive integers or both. We resolve three problems on asymptotic bases of order 2 which had been posed by Nathanson.