Two problems on subset sums

Abstract

For a set A of positive integers, let P(A) 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={b1<b2<⋯} is a sequence of integers with b1≥11, 3b1+5≤b2≤4b1, 3b2+2≤b3≤3b2+b1 and 3bn−bn−2≤bn+1≤3bn(n≥3), then there exists a set of positive integers A for which P(A)=N∖B. We also partially answer a problem of Wu by determining the structure of B={b1<b2<⋯} with b1>10 and b2>3b1+4, for which there exists a set of positive integers A such that P(A∩[0,bk])=[0,2bk]∖{bi,2bk−bi:1≤i≤k}(k≥2).