On finite nonnegative integer sets with identical representation functions

Let \(\mathbb {N}\) $\mathbb{N}$ be the set of all nonnegative integers. For \(S\subseteq \mathbb {N}\) $S\subseteq \mathbb{N}$ and \(n\in \mathbb {N}\) $n\in \mathbb{N}$ , let the representation function \(R_{S}(n)\) $R_S(n)$ denote the number of solutions of the equation \(n=s+s'\) $n=s+s^{\prime }$ with \(s, s'\in S\) $s,s^{\prime }\in S$ and \(s<s'\) $s<s^{\prime }$ . In this paper, we determine the structure of \(C, D\subseteq \mathbb {N}\) $C,D\subseteq \mathbb{N}$ with \(C\cup D=[0, m]\) $C\cup D=[0,m]$ , \(C\cap D=\{r_{1}, r_{2}\}\) $C\cap D=\{r_1,r_2\}$ , \(r_{1}<r_{2}\) $r_1<r_2$ and \(2\not \mid r_{1}\) $2\nmid r_1$ such that \(R_{C}(n)=R_{D}(n)\) $R_C(n)=R_D(n)$ for any nonnegative integer n.