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

TL;DR

This paper proves a conjecture about the possible orders of bases in finite cyclic groups, showing a structured relationship between the basis order and the group size, using additive number theory techniques.

Contribution

It establishes a bound on the order of bases in cyclic groups relative to the group size, confirming a conjecture and linking basis order to sumset growth.

Findings

For each k, there exists c_k such that basis orders are close to n/l for some l in [1,k]

The order of a basis exceeding n/k is within c_k of n/l for some integer l

The proof utilizes additive number theory results on sumset growth

Abstract

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.