WEIGHTED REPRESENTATION FUNCTIONS ON $\mathbb{Z}_m$

Abstract

Let $m$, $k_1$, and $k_2$ be three integers with $m\ge 2$. For $A\subseteq \mathbb{Z}_m$ and $n \in \mathbb{Z}_m$, let $\hat{r}_{k_1,k_2}(A,n)$ denote the number of solutions of the equation $n=k_1a_1+k_2a_2$ with $a_1,a_2\in A$. In this paper, we characterize all $m$, $k_1$, $k_2$, and $A$ for which $\hat{r}_{k_1,k_2}(\mathbb{Z}_m \setminus A,n) = \hat{r}_{k_1,k_2}(A,n)$ for all $n\in \mathbb{Z}_m$. As a corollary, we prove that there exists $A\subseteq \mathbb{Z}_m$ such that $\hat{r}_{k_1,k_2}(\mathbb{Z}_m\setminus A,n)=\hat{r}_{k_1,k_2}(A,n)$ for all $n\in \mathbb{Z}_m$ if and only if $2d \mid m$, where $d=(k_1,m)(k_2,m)/(k_1,k_2,m)^2$. We also pose several problems for further research.