The subsums of zero-sum free sequences in finite cyclic groups

Abstract

Let $\mathbb Z_n$ be the cyclic group of order $n \ge 3$ additively written. S. Savchev \& F. Chen (2007) proved that for each zero-sum free sequence $S = a_1 \bullet \dots \bullet a_t$ over $\mathbb Z_n$ of length $t > n/2$, there is an integer $g$ coprime to $n$ such that, if $\overline{r}$ denotes the least positive integer in the congruence class $r$ modulo $n$, then $\sum_{i=1}^t \overline{ga_i} < n$. Under the same hypothesis, in this paper we show that $$\left\{ \sum_{i \in Λ} \overline{ga_i} \;\; \Bigg| \;\; Λ\subset \{1,2,\dots,t\} \right\} = \left\{ 1, 2, \dots, \sum_{i=1}^t \overline{ga_i}\right\}.$$ It simplifies many calculations on inverse zero-sum problems.