On the integer sets with the same representation functions

Abstract

Let $\mathbb{N}$ be the set of all nonnegative integers. For $S\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let $R_S(n)$ denote the number of solutions of the equation $n=s_1+s_2$, $s_1,s_2\in S$ and $s_1<s_2$. Let $A$ be the set of all nonnegative integers which contain an even number of digits $1$ in their binary representations and $B=\mathbb{N}\setminus A$. Put $A_l=A\cap [0,2^l-1]$ and $B_l=B\cap [0,2^l-1]$. In 2017, Kiss and S\'{a}ndor proved that, if $C\cup D=[0,m]$, $0\in C$ and $C\cap D=\{r\}$, then $R_C(n)=R_D(n)$ for every positive integer $n$ if and only if there exists an integer $l\ge 1$ such that $r=2^{2l}-1$, $m=2^{2l+1}-2$, $C=A_{2l}\cup (2^{2l}-1+B_{2l})$ and $D=B_{2l}\cup (2^{2l}-1+A_{2l})$. This solved a problem of Chen and Lev. In this paper, we prove that, if $C \cup D=[0, m]\setminus \{r\}$ with $0<r<m$, $C \cap D=\emptyset$ and $0 \in C$, then $R_{C}(n)=R_{D}(n)$ for any nonnegative integer $n$ if and only if there exists an integer $l \geq 2$ such that $m=2^{l}$, $r=2^{l-1}$, $C=A_{l-1} \cup\left(2^{l-1}+1+B_{l-1}\right)$ and $D=B_{l-1} \cup\left(2^{l-1}+1+A_{l-1}\right)$.