Inverse problems for certain subsequence sums in integers

Let A be a nonempty finite set of k integers. Given a subset B of A, the sum of all elements of B is called the subset sum of B and we denote it by s(B). For a nonnegative integer a (<= k), let Sigma(alpha)(A) := {s(B) : B subset of A, |B| >= alpha}. Now, let A = ({a(1), ... , a1, (r1 copies), a(2), ... , a(2) (r2 copies), ... , a(k), ... , a(k) (rk copies)) be a finite sequence of integers with k distinct terms, where r(i) >= 1 for i = 1, 2, ... , k. Given a subsequence B of A, the sum of all terms of B is called the subsequence sum of B and we denote it by s(B). For 0 <= alpha <= Sigma(k)(i=1) r(i), let Sigma(alpha)(A) := {s(B) : B is a subsequence of A of length >= alpha}. Very recently, Balandraud obtained the minimum cardinality of Sigma(alpha)(A) in finite fields. Motivated by Balandraud's work, we find the minimum cardinality of Sigma(alpha)(A) in the group of integers. We also determine the structure of the finite set A of integers for which |Sigma(alpha)(A)| is minimal. Furthermore, we generalize these results of subset sums to the subsequence sums Sigma(alpha)(A). As special cases of our results, we obtain some already known results for the usual subset and subsequence sums. (c) 2020 Elsevier B.V. All rights reserved.