The extent to which subsets are additively closed (Q1024343)

In subset sum theory, sum-free sets have been the subject of many studies. In this paper, the authors consider the \(r\)-closed set, and in this case, the sum-free set is \(0\)-closed. An \(r\)-closed set is a comprehensive generalization of a sum-free set. There are many problems that can be considered on an \(r\)-closed set according to two different points of view, direct problem and inverse problem. In this paper, the authors investigate direct problems. They obtain some results on the combinatorial constant \(r(S)\), and in particular, the result obtained in cyclic groups of prime order can be considered as a generalization of the Cauchy-Davenport theorem.