The inverse problem on subset sums

For a set A , let P ( A ) be the set of all finite subset sums of A . We prove that if a sequence B = { b 1 < b 2 < ⋯ } of integers satisfies b 1 ≥ 11 , b 2 ≥ 3 b 1 + 5 , b 3 ≥ 3 b 2 + 3 and b n + 1 > 3 b n − b n − 2   ( n ≥ 3 ) , then there exists a sequence of positive integers A = { a 1 < a 2 < ⋯ } such that P ( A ) = N ∖ B . These lower bounds are optimal in a sense. We pose a problem for further research.