On an inverse problem in additive number theory

Abstract

Let A be a sequence of positive integers and P(A) be the set of all integers which can be represented as the finite sum of distinct terms of A. By improving a result of Hegyvári, Chen and Fang [2] proved that, for a sequence of integers ${B = \{b_{1} < b_{2} < \cdots \}}$ , if ${b_{1} \in \{4, 7, 8\} \cup \{b : b \geq 11\}}$ and ${b_{n+1} \geq 3b_{n} + 5}$ for all ${n \geq 1}$ , then there exists an infinite sequence A of positive integers for which ${P(A) = \mathbb{N} \setminus B}$ ; on the other hand, if b₂ =  3b₁ +  4, then such A does not exist. In this paper, for b₂ = 3b₁ +  5, we determine the critical value for b₃ such that there exists an infinite sequence A of positive integers for which ${P(A) = \mathbb{N} \setminus B}$ .