  We prove a conjecture of Dukes and Herke concerning the possible orders of a
basis for the cyclic group Z_n, namely : For each k \in N there exists a
constant c_k > 0 such that, for all n \in N, if A \subseteq Z_n is a basis of
order greater than n/k, then the order of A is within c_k of n/l for some
integer l \in [1,k]. The proof makes use of various results in additive number
theory concerning the growth of sumsets.
