Let A = (a0, . . . , a0 | {z } r copies , a1, . . . , a1 | {z } r copies , . . . , ak1, . . . , ak1 | {z } r copies ) be a finite sequence of integers, where a0 < a1 < < ak1, k 1 and r 1. Given a subsequence, the sum of all the terms of the subsequence is called the subsequence sum. The set of all nontrivial subsequence sums of A is denoted by S(r, A), where A = {a0, a1, . . . , ak1} is the set of distinct terms of the sequence A, called the associated set of the sequence A. For r = 1, this sumset is the usual sumset S(A) of nontrivial subet sums of A. The direct problem for the sumset S(r, A) is to find a lower bound for |S(r, A)| in terms of |A| and r. The inverse problem for S(r, A) is to determine the structure of the finite set A of integers for which |S(r, A)| is minimal. In this paper, we give new proofs of existing direct and inverse theorems for S(r, A) using the direct and inverse theorems of Nathanson for S(A).
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