paper

On an inverse problem in additive number theory

arXiv:2005.09186

Abstract

For a set $A$ , let $P (A)$ be the set of all finite subset sums of $A$ . In this paper, for a sequence of integers $B = {1 < b_{1} < b_{2} < \dots}$ and $3 b_{1} + 5 \leq b_{2} \leq 6 b_{1} + 10$ , we determine the critical value for $b_{3}$ such that there exists an infinite sequence $A$ of positive integers for which $P (A) = N ∖ B$ . This result shows that we partially solve the problem of Fang and Fang [`On an inverse problem in additive number theory', Acta Math. Hungar. 158(2019), 36-39].