On the possible orders of a basis for a finite cyclic group

Journal article, 2010

We prove a result concerning the possible orders of a basis for the cyclic group Z(n), namely: For each k is an element of N there exists a constant c(k) > 0 such that, for all n is an element of N, if A subset of 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 is an element of [1, k]. The proof makes use of various results in additive number theory concerning the growth of sumsets. Additionally, exact results are summarized for the possible basis orders greater than n/4 and less than root n. An equivalent problem in graph theory is discussed, with applications.