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 representation function \(R_{A}(\overline{n})\) denote the number of solutions of the equation \(\overline{n}=\overline{a}+\overline{a'}\) with ordered pairs \((\overline{a}, \overline{a'})\in A \times A\). In this paper, we determine all sets \(A, B\subseteq \mathbb {Z}_{m}\) with \(A\cup B=\mathbb {Z}_{m}\) and \(|A\cap B|=2\) or \(m-2\) such that \(R_{A}(\overline{n})=R_{B}(\overline{n})\) for all \(\overline{n}\in \mathbb {Z}_{m}\). We also prove that if m is a positive integer with \(4\mid m\), then there exist two distinct sets \(A, B\subseteq \mathbb {Z}_{m}\) with \(A\cup B=\mathbb {Z}_{m}\) and \(|A\cap B|=4\) or \(m-4\), \(B\ne A+\overline{\frac{m}{2}}\) such that \(R_{A}(\overline{n})=R_{B}(\overline{n})\) for all \(\overline{n}\in \mathbb {Z}_{m}\). If m is a positive integer with \(m\equiv 2\pmod 4\), \(A\cup B=\mathbb {Z}_{m}\) and \(|A\cap B|=4\) or \(m-4\), then \(R_{A}(\overline{n})=R_{B}(\overline{n})\) for all \(\overline{n}\in \mathbb {Z}_{m}\) if and only if \(B=A+\overline{\frac{m}{2}}\).

对于任何正整数m，令\(\mathbb {Z}_{m}\)为模m的残差类集合。对于\(A\subseteq \mathbb {Z}_{m}\)和\(\overline{n}\in \mathbb {Z}_{m}\)，让表示函数\(R_{A}(\ overline{n})\)表示方程\(\overline{n}=\overline{a}+\overline{a'}\)与有序对\((\overline{a}, \) 的解数overline{a'})\in A \times A\)。在本文中，我们确定所有集合\（A，B \ subseteq \ mathbb {Z} _ {米} \）与\（A \杯B = \ mathbb {Z} _ {米} \）和\（| A \cap B|=2\)或\(m-2\)使得\(R_{A}(\overline{n})=R_{B}(\overline{n})\)对于所有\(\overline{n}\in \mathbb {Z}_{m}\)。我们还证明，如果m是一个带有\(4\mid m\)的正整数，那么存在两个不同的集合\(A, B\subseteq \mathbb {Z}_{m}\)和\(A\cup B=\mathbb {Z}_{m}\)和\(|A\cap B|=4\)或\(m-4\) , \(B\ne A+\overline{\frac{m}{ 2}}\)使得\(R_{A}(\overline{n})=R_{B}(\overline{n})\)对于所有\(\overline{n}\in \mathbb {Z} _{m}\)。如果m是具有\(m\equiv 2\pmod 4\)的正整数，则\(A\cup B=\mathbb {Z}_{m}\)和\(|A\cap B|=4\)或\(m-4\)，然后\(R_{A}(\overline{n})=R_{B}(\overline{n})\)对于所有\(\overline{n}\in \mathbb {Z}_{m}\)当且仅当\(B=A+\overline{\frac{m}{2}}\)。