On an inverse problem in additive number theory

TL;DR

This paper investigates an inverse problem in additive number theory, identifying conditions under which a sequence of integers can be constructed so that its finite subset sums match all natural numbers except for a specified sequence.

Contribution

It determines the critical value of the third term in a sequence to ensure the existence of an infinite set with prescribed subset sum properties, advancing the understanding of inverse problems in additive number theory.

Findings

Identifies the critical value for the third term in the sequence.

Constructs an infinite sequence with prescribed subset sum complement.

Partially solves a problem posed by Fang and Fang.

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 $B=\{1<b_1<b_2<\cdots\}$ and $3b_1+5\leq b_2\leq 6b_1+10$, we determine the critical value for $b_3$ such that there exists an infinite sequence $A$ of positive integers for which $P(A)=\mathbb{N}\setminus B$. 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].