Let \(\mathbb {N}\) denote the set of positive integers. For a set \(A\subseteq \mathbb {N}\) , we write \(A\sim \mathbb {N}\) if and only if A contains all sufficiently large integers. For a positive integer j and a set \(B\subseteq \mathbb {N}\) , we denote by jB the set of all integers that can be expressed as the sum of j elements of B (allowing repeats), and by \(j\times B\) the set of all integers that can be expressed as the sum of exactly j distinct elements of B. A set B is called an asymptotic basis if there exists a positive integer h such that \(B\cup 2B\cup \cdots \cup hB\sim \mathbb {N}\) . The order of an asymptotic basis B is the smallest positive integer h for which \(\bigcup _{j=1}^{h} (jB) \sim \mathbb {N}\) . Similarly, the restricted order of an asymptotic basis B is the smallest positive integer h for which \(\bigcup _{j=1}^{h} (j \times B) \sim \mathbb {N}\) . In this paper, we prove that if B is a set of positive integers satisfying \(n/2\le B(n)\le n/2+k\) for all positive integers n, then \(B\cup (2\times B)\sim \mathbb {N}\) , where k is a positive integer. Furthermore, we also show that both the upper bound and the lower bound are optimal.
