1 Introduction

Let $\mathbb {N}$ be the set of all nonnegative integers. For a set $A\subseteq \mathbb {N}$ , define the additive representation function

$$ \begin{align*} R_1(A,n)=|\{(a_1,a_2): n=a_1+a_2,~a_1,a_2\in A\}| \end{align*} $$

and the ordered representation functions

$$ \begin{align*} R_2(A,n)& =|\{(a_1,a_2): n=a_1+a_2,~a_1,a_2\in A,~a_1<a_2\}|,\\ R_3(A,n)& =|\{(a_1,a_2): n=a_1+a_2,~a_1,a_2\in A,~a_1\le a_2\}|. \end{align*} $$

For each $i\in \{1,2,3\}$ , Sárközy asked whether there exist two sets $A,B\subseteq \mathbb {N}$ with infinite symmetric difference, that is,

$$ \begin{align*}|(A\cup B)\setminus(A\cap B)|=\infty\end{align*} $$

and

$$ \begin{align*}R_i(A,n)=R_i(B,n)\end{align*} $$

for all sufficiently large integers n. For $i=1$ , Dombi [Reference Dombi3] observed that $R_1(A, 2m)$ is odd if and only if $m\in A$ . So $R_1(A, 2m)\neq R_1(B, 2m)$ for all integers $m\in (A\cup B)\setminus (A\cap B),$ and the answer is negative for $i=1$ . In 2002, Dombi [Reference Dombi3] proved that the answer is positive for $i=2$ . In 2003, Chen and Wang [Reference Chen and Wang2] proved that the answer is also positive for $i=3$ . Later, many authors studied this problem by means of many different methods. In particular, Sándor [Reference Sándor5] and Tang [Reference Tang6] proved the following nice results in different ways.

From Dombi’s discussion, it can be seen that the reason why the answer for $i=1$ is negative is because the parity of $R_1(A,n)$ and $R_1(\mathbb {N}\setminus A,n)$ is different when n is even. The authors of this paper and Cheng [Reference Yan, Li, Li and Cheng9] avoided the case where n is even, and they determined the structure of the set $A\subseteq \mathbb {N}$ such that $R_1(A,2k-1)=R_1(\mathbb {N}\setminus A,2k-1)$ for all integers $k\ge K$ . In addition, they obtained the following results.

Theorem 1.3 [Reference Yan, Li, Li and Cheng9].

Let $A\subseteq \mathbb {N}$ , K be a positive integer and let g be an even integer with $-2K\le g\le 2K$ . The equations

$$ \begin{align*}R_1(A,2k-1)-R_1(\mathbb{N}\setminus A,2k-1)=g,~~R_1(A,2k)-R_1(\mathbb{N}\setminus A,2k)=g+1\end{align*} $$

hold for all integers $k\ge K$ if and only if

$$ \begin{align*} A=A_0\cup\{2k: k\ge K\}, \end{align*} $$

where $A_0\subseteq [0,2K-1]$ and $|A_0|=K+g/2$ .

Theorem 1.4 [Reference Yan, Li, Li and Cheng9].

Let $A\subseteq \mathbb {N}$ , let K be a nonnegative integer and let g be an odd integer with $-2K-1\le g\le 2K+1$ . The equations

$$ \begin{align*}R_1(A,2k)-R_1(\mathbb{N}\setminus A,2k)=g,~~R_1(A,2k+1)-R_1(\mathbb{N}\setminus A,2k+1)=g+1\end{align*} $$

hold for all integers $k\ge K$ if and only if

$$ \begin{align*} A=A_0\cup\{2k+1: k\ge K\}, \end{align*} $$

where $A_0\subseteq [0,2K]$ and $|A_0|=K+(g+1)/2$ .

There are many results about the ordered representation functions $R_2(A,n)$ and $R_3(A,n)$ . Cao-Labora et al. [Reference Cao-Labora, Rué and Spiegel1] showed that the ordered representation functions satisfy an Erdős–Fuchs-type result. Tang [Reference Tang7] considered Problem 6 of Erdős et al. [Reference Erdős, Sárközy and Sós4] about locally small and locally large values of the ordered representation functions. In this paper, we consider the following problem related to ordered representation functions.

Problem 1.5. Let $g_1,g_2$ be nonnegative integers with $g_1\neq g_2$ . For $i\in \{2,3\}$ , do there exist infinite sets $A\subseteq \mathbb {N}$ and $T\subseteq \mathbb {N}$ such that

$$ \begin{align*}R_i(A,n)-R_i(\mathbb{N}\setminus A,n)=g_1\quad \text{for all }n\in T\end{align*} $$

and

$$ \begin{align*}R_i(A,n)-R_i(\mathbb{N}\setminus A,n)=g_2\quad \text{for all }n\in \mathbb{N}\setminus T\end{align*} $$

for all sufficiently large integers n?

If $-N\le g_1=g_2\le N$ , the structure of a set $A\subseteq \mathbb {N}$ such that $R_i(A,n)-R_i(\mathbb {N}\setminus A,n)=g_1$ for all integers $n\ge 2N-1$ was provided in [Reference Yan8], where N is a positive integer. In this paper, we prove that if $g_1\neq g_2$ and $T=2\mathbb {N}$ , then such a set A in Problem 1.5 does not exist.

Theorem 1.6. Let $g_1,g_2$ be nonnegative integers with $g_1\neq g_2$ . Then there does not exist any set $A\subseteq \mathbb {N}$ such that

$$ \begin{align*}R_2(A,2k)-R_2(\mathbb{N}\setminus A,2k)=g_1,~~R_2(A,2k+1)-R_2(\mathbb{N}\setminus A,2k+1)=g_2\end{align*} $$

for all sufficiently large integers k.

Theorem 1.7. Let $g_1,g_2$ be nonnegative integers with $g_1\neq g_2$ . Then there does not exist any set $A\subseteq \mathbb {N}$ such that

$$ \begin{align*}R_3(A,2k)-R_3(\mathbb{N}\setminus A,2k)=g_1,~~R_3(A,2k+1)-R_3(\mathbb{N}\setminus A,2k+1)=g_2\end{align*} $$

for all sufficiently large integers k.