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,$n..S=\{s_1+\cdots +s_n\mid s_1,\dots ,s_n\in S\}.$ We say that $S$ is an $(n,0)$-set if $0\notin n..S$. Let $k,s$ be integers with $k\ge 2$ 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.