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

TL;DR

This paper characterizes conditions under which weighted representation functions on modular integers are symmetric with respect to a subset and its complement, using exponential sums.

Contribution

It provides a complete characterization of when weighted representation counts are equal for a subset and its complement in modular arithmetic.

Findings

Characterization of all parameters where representation functions are equal

Use of exponential sums to analyze representation functions

Identification of open problems for future research

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 $n=k_1a_1+k_2a_2$ 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.