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2013 WEIGHTED REPRESENTATION FUNCTIONS ON $\mathbb{Z}_m$
Quan-Hui Yang, Yong-Gao Chen
Taiwanese J. Math. 17(4): 1311-1319 (2013). DOI: 10.11650/tjm.17.2013.2463

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.

Citation

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Quan-Hui Yang. Yong-Gao Chen. "WEIGHTED REPRESENTATION FUNCTIONS ON $\mathbb{Z}_m$." Taiwanese J. Math. 17 (4) 1311 - 1319, 2013. https://doi.org/10.11650/tjm.17.2013.2463

Information

Published: 2013
First available in Project Euclid: 10 July 2017

zbMATH: 1357.11016
MathSciNet: MR3085513
Digital Object Identifier: 10.11650/tjm.17.2013.2463

Subjects:
Primary: 11B34, 11L03

Rights: Copyright © 2013 The Mathematical Society of the Republic of China

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Vol.17 • No. 4 • 2013
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