Weighted representation functions on ${\mathbb{Z}}_m$

Abstract

Let $m$, $k_1$, and $k_2$ be three integers with $m\ge 2$. For any set $A\subseteq {\mathbb{Z}}_m$ and $n\in {\mathbb{Z}}_m$, let ${\hat{r}}_{k_1,k_2}(A,n)$ denote the number of solutions of the equation $n=k_1{a}_1+k_2{a}_2$ with $a_1,a_2\in A$. In this paper, using exponential sums, we characterize all $m$, $k_1$, $k_2$, and $A$ for which ${\hat{r}}_{k_1,k_2}(A,n)={\hat{r}}_{k_1,k_2}({\mathbb{Z}}_m\setminus A,n)$ for all $n\in {\mathbb{Z}}_m$. We also pose several problems for further research.

9 pages