On a problem in additive number theory

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={b1<b2<⋯} of integers satisfies b1≧11 and bn+1≧3bn+5 (n=1,2,…), then there exists a sequence of positive integers A={a1<a2<⋯} for which P(A)=ℕ∖B. On the other hand, if a sequence B={b1<b2<⋯} of positive integers satisfies either b1=10 or b2=3b1+4, then there is no sequence A of positive integers for which P(A)=ℕ∖B.