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 Hegyvari, 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 $${P(A) = \mathbb{N} \setminus B}$$ .