1 Introduction
Let $\mathbb {N}$ be the set of all nonnegative integers. For a set $A\subseteq \mathbb {N}$ , let $R_{1}(A,n)$ , $R_{2}(A,n)$ and $R_{3}(A,n)$ denote the number of solutions of $a_1+a_2=n, a_1,a_2\in A$ ; $a_1+a_2=n, a_1, a_2\in A, a_1<a_2$ and $a_1+a_2=n, a_1,a_2\in A, a_1\leq a_2$ , respectively. For $i=1,2,3$ , Sárközy asked whether there exist two sets A and B with $|(A\cup B)\setminus (A\cap B)|=+\infty $ such that $R_{i}(A,n)=R_{i}(B,n)$ for all sufficiently large integers n. We call this problem the Sárközy problem. In 2002, Dombi [Reference Dombi2] proved that the answer is negative for $i=1$ and positive for $i=2$ . For $i=3$ , Chen and Wang [Reference Chen and Wang1] proved that the answer is also positive. In 2004, Lev [Reference Lev3] provided a new proof by using generating functions. Later, Sándor [Reference Sándor5] determined the partitions of $\mathbb {N}$ into two sets with the same representation functions by using generating functions. In 2008, Tang [Reference Tang6] provided a simple proof by using the characteristic function.
In 2012, Yang and Chen [Reference Yang and Chen7] first considered the Sárközy problem with weighted representation functions. For any positive integers $k_1,\ldots ,k_t$ and any set $A\subseteq \mathbb {N}$ , let $R_{k_1,\ldots ,k_t}(A,n)$ be the number of solutions of the equation $n=k_1a_1+\cdots +k_ta_t$ with $a_1,\ldots ,a_t\in A$ . They posed the following question.
Problem 1.1 [Reference Yang and Chen7, Problem 1]
Does there exist a set $A\subseteq \mathbb {N}$ such that $R_{k_1,\ldots ,k_t}(A,n)=R_{k_1,\ldots ,k_t}(\mathbb {N}\setminus A,n)$ for all $n\geq n_0$ ?
They answered this question for $t=2$ and proved the following results.
Theorem 1.2 [Reference Yang and Chen7, Theorem 1].
If $k_1$ and $k_2$ are two integers with $k_2>k_1\geq 2$ and $(k_1,k_2)=1$ , then there does not exist any set $A\subseteq \mathbb {N}$ such that $R_{k_1,k_2}(A,n)=R_{k_1,k_2} (\mathbb {N}\setminus A,n)$ for all sufficiently large integers n.
Theorem 1.3 [Reference Yang and Chen7, Theorem 2].
If k is an integer with $k>1$ , then there exists a set $A\subseteq \mathbb {N}$ such that
for all integers $n\geq 1$ .
Furthermore, if $0\in A$ , then (1.1) holds for all integers $n\geq 1$ if and only if
where $[x,y]=\{n:n\in \mathbb {Z},x\leq n\leq y\}$ .
Later, Li and Ma [Reference Li and Ma4] proved the same results by using generating functions.
Let g be a fixed integer. In this paper, we consider whether there exists a set $A\subseteq \mathbb {N}$ such that $R_{k_1,k_2}(A,n)-R_{k_1,k_2}(\mathbb {N}\setminus A,n)=g$ for all $n\geq n_0$ . First, we answer this problem in the negative if $k_1$ and $k_2$ are two integers with $2\le k_1<k_2$ and $(k_1,k_2)=1$ .
Theorem 1.4. Let g be a fixed integer. If $k_1$ and $k_2$ are two integers with $2\le k_1<k_2$ and $(k_1,k_2)=1$ , then there does not exist any set $A\subseteq \mathbb {N}$ such that
for all sufficiently large integers n.
Similar to Theorem 1.3, we seek a set $A\subseteq \mathbb {N}$ such that $R_{1,k}(A,n)-R_{1,k}(\mathbb {N}\setminus A,n)=g$ for all integers $n\geq 1$ . In fact, if $|g|>1$ , then such a set A does not exist by the simple observation that $0\le R_{1,k}(A,n)\le 1$ and $0\le R_{1,k}(\mathbb {N}\setminus A,n)\le 1$ for all positive integers $n<k$ . So we only need to consider the case $g=1$ .
Theorem 1.5. If k is an integer with $k>1$ , then there exists a set $A\subseteq \mathbb {N}$ such that
for all integers $n\ge 1$ .
Furthermore, (1.2) holds for all integers $n\geq 1$ if and only if
2 Proofs
Lemma 2.1. Let $k_1<k_2$ be two positive integers, $\{a(n)\}_{n=-\infty }^{+\infty }$ be a sequence of integers with $a(n)=0$ for $n<0$ and $A\subseteq \mathbb {N}$ . Then the equality
holds for all nonnegative integers n if and only if
holds for all nonnegative integers n, where $\chi _A(i)$ is the characteristic function of A, that is, $\chi _A(i) = 1$ if $i\in A$ and $\chi _A(i) = 0$ if $i\notin A$ .
Proof. Let $f(x)$ be the generating function associated with A, that is,
Then,
Let
It follows that (2.1) holds for all nonnegative integers n if and only if
that is,
Note that
and
It follows from (2.2) that for all nonnegative integers n,
This completes the proof of Lemma 2.1.
Lemma 2.2. Let $n_0$ be a positive integer and $k_1<k_2$ be two positive integers with $(k_1,k_2)=1$ and $A\subseteq \mathbb {N}$ be a set with
If $n\ge k_1+k_2+n_0$ and $\chi _{A}(n)+\chi _{A}(n+1)=1$ , then $k_2\mid n+1$ .
Proof. Since $\chi _{A}(n)+\chi _{A}(n+1)=1$ , it follows that
By (2.3),
and
It follows from (2.4) that
Let t and r be integers with
If $r\ge 1$ , then
which is a contradiction. Hence, $r=0$ and $(n+1)k_1=tk_2$ . Noting that $(k_1,k_2)=1$ , we have $k_2\mid n+1$ . This completes the proof of Lemma 2.2.
Proof of Theorem 1.4.
Let g be an integer and let $k_1,k_2$ be integers with $2\le k_1<k_2$ and $(k_1,k_2)=1$ . Suppose that
for all integers $n\ge n_0$ . Let $\{a(n)\}_{n=-\infty }^{+\infty }$ be a sequence of integers with $a(n)=0$ for $n<0$ and $a(n)=g$ for all integers $n\ge n_0$ . It follows from Lemma 2.1 that for all integers $i\ge k_1+k_2+n_0$ ,
If A is a finite set, then $R_{k_1,k_2}(A,n)=0$ for all sufficiently large integers n, and $R_{k_1,k_2} (\mathbb {N}\setminus A,n)$ cannot be a fixed constant as $n\rightarrow +\infty $ , which implies that (2.5) cannot hold. So A is an infinite set. Similarly, $\mathbb {N}\setminus A$ is also an infinite set.
Since $2\le k_1<k_2$ , it follows that there exists an integer $t>1$ such that $k_2< k_1^{t}$ . Note that both A and $\mathbb {N}\setminus A$ are infinite sets. So there exists an integer $n=k_1^{\alpha }k_2^{\beta }h-1>(k_1+k_2+n_0)^{t+1}$ such that $n\in A$ and $n+1\notin A$ , where $\alpha $ and $\beta $ are nonnegative integers and h is a positive integer with $(h,k_1k_2)=1$ . It follows from (2.6) and Lemma 2.2 that $k_2\mid n+1$ and $\beta \ge 1$ . Since
it follows that $k_1^{\alpha +\beta }>k_1+k_2+n_0$ or $h>k_1+k_2+n_0$ . Hence, for any $0\le i\le \beta $ ,
By (2.6),
and
Since $k_1^{\alpha }k_2^{\beta }h=n+1\notin A$ and $k_1^{\alpha }k_2^{\beta }h-1=n\in A$ , it follows from (2.8) and (2.9) that
By Lemma 2.2, $k_2\mid k_1^{\alpha +1}k_2^{\beta -1}h$ and so $\beta \ge 2$ . Continuing this procedure yields
By (2.7) and Lemma 2.2, we also have $k_2\mid k_1^{\alpha +\beta }h$ , which is impossible. Hence, there does not exist any set $A\subseteq \mathbb {N}$ such that (2.5) holds for all sufficiently large integers n. This completes the proof of Theorem 1.4.
Proof of Theorem 1.5.
Suppose that there is a set A such that
for all integers $n\ge 1$ . Then $0 \in A$ and (2.10) holds for all integers $n\ge 0$ . Let $\{a(n)\}_{n=-\infty }^{+\infty }$ be a sequence of integers with $a(n)=0$ for $n<0$ and $a(n)=1$ for $n\ge 0$ . By Lemma 2.1,
for all nonnegative integers n if and only if
for all nonnegative integers n, that is,
Thus,