On subsequence sums of a zero-sum free sequence over finite abelian groups. (English) Zbl 1469.11039
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 \(1<n_1\mid \ldots\mid n_r\). The authors also showed that if \(n_{r-1}\ge 5\), then \(\mathsf f_G(n_r+3)\ge 5n_r-1\) (see Theorem 1.5).
Reviewer: Qinghai Zhong (Graz)
MSC:
| 11B75 | Other combinatorial number theory |
| 11P70 | Inverse problems of additive number theory, including sumsets |
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