On the difference bases of $$\mathbb {Z}_{m}$$

Abstract

For any positive integer m, let \(\mathbb {Z}_m\) be the cyclic group of order m. For any subset \(A\subseteq \mathbb {Z}_{m}\) and any \(n\in \mathbb {Z}_{m}\), let \(\delta _{A}(n)=\#\{(a,b)|n=a-b, a\in A, b\in A\}\). In this paper, we prove that, for any positive integer m, there exists a subset A of \(\mathbb {Z}_m\) such that \(\delta _A (n)\le 5\) for all \(n \in \mathbb {Z}_m\) with at most 3 exceptions, which improves a 2010 result of Y.–G. Chen & T. Sun.