On partitions of $\mathbb{Z}_{m}$ with the same representation function

Abstract: For any positive integer $m$, let $\mathbb{Z}{m}$ be the set of residue classes modulo $m$. For $A\subseteq \mathbb{Z}{m}$ and $\overline{n}\in \mathbb{Z}{m}$, let $R{A}(\overline{n})$ denote the number of solutions of $\overline{n}=\overline{a}+\overline{a'}$ with unordered pairs $(\overline{a}, \overline{a'})\in A \times A$. In this paper, we prove that if $m=2^{\alpha}$ with $\alpha\neq 2$, $A\cup B=\mathbb{Z}{m}$ and $|A\cap B|=2$, then $R{A}(\overline{n})=R_{A}(\overline{n})$ for all $\overline{n}\in \mathbb{Z}_{m}$ if and only if $B=A+\overline{\frac{m}{2}}$.