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 ∈ F 2 f , where F ∈ F ∗ ( W ) . Let N = W 1 ∪ W 2 be a partition with 0 ∈ W 1 such that W 1 and W 2 are infinite. In this paper, we prove that A = A ( W 1 ) ∪ A ( W 2 ) is a minimal asymptotic basis of order 2 if and only if either W 1 contains no consecutive integers or W 2 contains consecutive integers or both. We resolve three problems on asymptotic bases of order 2 which had been posed by Nathanson.