On a subset sums problem of Chen and Wu

TL;DR

This paper constructs a specific sequence of positive integers whose subset sums cover all natural numbers except a particular sequence, providing a negative answer to an inverse subset sum problem posed by Chen and Wu.

Contribution

It demonstrates the existence of a sequence with a prescribed complement in its subset sums, resolving an open problem by Chen and Wu.

Findings

Constructed a sequence with a specific subset sum complement

Showed the complement sequence satisfies given recurrence relations

Provided a counterexample to the inverse problem on subset sums

Abstract

For a set $A$, let $P(A)$ be the set of all finite subset sums of $A$. We prove that if a sequence $B=\{11\leq b_1<b_2<\cdots\}$ satisfies $b_2=3b_1+5$, $b_3=3b_2+2$ and $b_{n+1}=3b_n+4b_{n-1}$ for all $n\geq 3$, then there is a sequence of positive integers $A=\{a_1<a_2<\cdots\}$ such that $P(A)=\mathbb{N}\setminus B$. This result shows that the answer to the problem of Chen and Wu [`The inverse problem on subset sums', European. J. Combin. 34(2013), 841-845] is negative.