Some Properties of \((k,0)\)-Sets of Cyclic Groups

Abstract

Let \(S\) be a nonempty subset of the cyclic group \(\mathbb{Z}_p\), where \(p\) is an odd prime. Denote the \(n\)-fold sum of \(S\) as \(n..S\). That is,\(n..S = \{s_1 + \cdots + s_n \mid s_1, \ldots, s_n \in S\}.\) We say that \(S\) is an \((n, 0)\)-set if \(0 \notin n..S\). Let \(k, s\) be integers with \(k \geq 2\) such that \(p-1 = ks\). In this paper, we determine the number of \((k, 0)\)-sets of \(\mathbb{Z}_p\) which are in arithmetic progression and show explicitly the forms taken by those \((k, 0)\)-sets which achieve the maximum cardinality.