Exhaustion numbers of subsets of Abelian groups

Let G be a finite group written additively and S a non-empty subset of G. We say that S is e-exhaustive if G = S+...+S (e times). The minimal integer e > 0, if it exists, such that S is e-exhaustive, is called the exhaustion number of the set S and is denoted by e(S). In this paper we completely determine the exhaustion numbers of subsets of Abelian groups which are in arithmetic progression. The exhaustion numbers of various subsets of Abelian groups which are not in arithmetic progression are also determined.