Treffer: On subsequence sums of a zero-sum free sequence over finite abelian groups

Let \(G\) be a finite abelian group. A sequence over \(G\) is a finite unorder sequence with terms from \(G\) and repetition allowed. Let \(S=g_1\cdot\ldots\cdot g_{\ell}\) be a sequence over \(G\). We define \(\Sigma(S)=\{\sum_{i\in I}g_i\colon \emptyset\neq I\subset [1,\ell]\}\) and we say \(S\) is zero-sum free if \(0\not\in \Sigma(S)\). For every \(r\in \mathbb N\), we let \[ \mathsf f_G(r)=\min\big\{|\Sigma(S)|\colon S\text{ is a zero-sum free sequence over \(G\) of length }r\big\}\,. \] Under some mild conditions, the authors proved that \(|\Sigma(S)|\ge 5|S|-16\) for all zero-sum free sequences \(S\) over \(G\) of length \(|S\ge 5\) (see Theorem 1.1). Suppose \(G\cong C_{n_1}\oplus \ldots\oplus C_{n_r}\) with \(1