Inverse theorems for subset sums

Abstract:

Let $A$ be a finite set of integers. For $h \geqslant 1$, let ${S_h}(A)$ denote the set of all sums of $h$ distinct elements of $A$. Let $S(A)$ denote the set of all nonempty sums of distinct elements of $A$. The direct problem for subset sums is to find lower bounds for $|{S_h}(A)|$ and $|S(A)|$ in terms of $|A|$. The inverse problem for subset sums is to determine the structure of the extremal sets $A$ of integers for which $|{S_h}(A)|$ and $|S(A)|$ are minimal. In this paper both the direct and the inverse problem for subset sums are solved.