Representation functions in the set of natural numbers

Abstract

Let $N$ be the set of all nonnegative integers. For $S\subseteq N$ and $n\in N$, let $R_S\left(n\right)$ denote the number of solutions of the equation $n=s+s^{\prime }$, $s,s^{\prime }\in S$, $s<s^{\prime }$. In this paper, we determine the structure of all sets A and B such that $A\cup B=N\setminus \{r+mk:k\in N\}$, $A\cap B=\varnothing$ and $R_A\left(n\right)=R_B\left(n\right)$ for every positive integer n, where m and r are two integers with $m\ge 2$ and $r\ge 0$.