Inverse theorems for subset sums

Abstract

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