On the critical values in subset sum

Abstract

For a sequence A of positive integers, let P(A) be the set of all integers which can be represented as the finite sum of distinct terms of A. In 2012, Chen and Fang proved that, for a sequence of integers B={b1<b2<⋯}, if b1∈{4,7,8}∪{b:b≥11,b∈N} and bn+1≥3bn+5 for all n≥1, then there exists an infinite sequence A of positive integers for which P(A)=N∖B; on the other hand, if b2=3b1+4, then such A does not exist. Recently, for b2=3b1+5, the authors determined the critical value for b3 such that there exists an infinite sequence A of positive integers for which P(A)=N∖B. In this paper, we fix the exact critical values for the above general terms.