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

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