On the structure of non-negative integer sets which have identical representation functions

Abstract Let $\mathbb{N}$ be the set of all non-negative integers. For any integer r and m , let $r+m\mathbb{N}=\{r+mk: k\in\mathbb{N}\}$ . For $S\subseteq \mathbb{N}$ and $n\in \mathbb{N}$ , let $R_{S}(n)$ denote the number of solutions of the equation $n=s+s'$ with $s, s'\in S$ and $s \lt s'$ . Let $r_{1}, r_{2}, m$ be integers with $0 \lt r_{1} \lt r_{2} \lt m$ and $2\mid r_{1}$ . In this paper, we prove that there exist two sets C and D with $C\cup D=\mathbb{N}$ and $C\cap D=(r_{1}+m\mathbb{N})\cup (r_{2}+m\mathbb{N})$ such that $R_{C}(n)=R_{D}(n)$ for all $n\in\mathbb{N}$ if and only if there exists a positive integer l such that $r_{1}=2^{2l+1}-2, r_{2}=2^{2l+1}-1, m=2^{2l+2}-2$ .