Note on the Inverse Problems Associated with Subsequence Sums

Let \( G = C_n \oplus C_n \) with \( n \geq 3 \) and \( S \) be a sequence with elements of \( G \). Let \( \Sigma(S) \subseteq G \) denote the set of group elements which can be expressed as a sum of a nonempty subsequence of \( S \). In this note, we show that if \( S \) contains \( 2n – 3 \) elements of \( G \), then either \( 0 \in \Sigma(S) \) or \( |\Sigma(S)| \geq n^2 – n – 1 \). Moreover, we determine the structures of the sequence \( S \) over \( G \) with length \( |S| = 2n – 3 \) such that \( 0 \notin \Sigma(S) \) and \( |\Sigma(S)| = n^2 – n – 1 \).