The Inverse Problem on Subset Sums, II

For a set T of integers, let P(T) be the set of all finite subset sums of T, and let T(x) be the set of all integers of T not exceeding x. Let $B=\{b_{1}<b_{2}<\cdots \}$ be a sequence of integers and d₁=10, d₂=3b₁+4, and d_n=3b_n-1+2 $(n\ge 3)$ . In this paper, we prove that

(i) if b_n>d_n for all $n\ge 1$ , then there exists a sequence of positive integers $A=\{a_{1}<a_{2}<\cdots \}$ such that, for all $k\ge 2$ , $P(A(b_k))=[0,2b_k]\setminus \{b_u, 2b_k-b_u : 1\le u\le k\}$ ;

(ii) if b_m=d_m for some $m\ge 1$ and b_n>d_n for all $n\not= m$ , then there is no such sequence A.

We also pose a problem for further research.

Received May 7 2013; revised version received September 5 2013. Published in Journal of Integer Sequences, October 12 2013.

The Inverse Problem on Subset Sums, II

Jian-Dong Wu School of Mathematical Sciences and Institute of Mathematics Nanjing Normal University Nanjing 210023 P. R. China

Jian-Dong Wu
School of Mathematical Sciences and Institute of Mathematics
Nanjing Normal University
Nanjing 210023
P. R. China