Two problems on subset sums

Abstract

For a set $A$ of positive integers, let $P\left(A\right)$ denote the set of all finite subset sums of $A$. In this paper, we completely solve a problem of Chen and Wu by proving that if $B=\{b_1<b_2<\cdots \}$ is a sequence of integers with $b_1\ge 11$, $3b_1+5\le b_2\le 4b_1$, $3b_2+2\le b_3\le 3b_2+b_1$ and $3b_n-b_{n-2}\le b_{n+1}\le 3b_n(n\ge 3)$, then there exists a set of positive integers $A$ for which $P\left(A\right)=N\setminus B$. We also partially answer a problem of Wu by determining the structure of $B=\{b_1<b_2<\cdots \}$ with $b_1>10$ and $b_2>3b_1+4$, for which there exists a set of positive integers $A$ such that $P(A\cap [0,b_k])=[0,2b_k]\setminus \{b_i,2b_k-b_i:1\le i\le k\}(k\ge 2)$.