Taiwanese Journal of Mathematics

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

Quan-Hui Yang and Yong-Gao Chen

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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.

Article information

Source
Taiwanese J. Math., Volume 17, Number 4 (2013), 1311-1319.

Dates
First available in Project Euclid: 10 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499706119

Digital Object Identifier
doi:10.11650/tjm.17.2013.2463

Mathematical Reviews number (MathSciNet)
MR3085513

Zentralblatt MATH identifier
1357.11016

Subjects
Primary: 11B34: Representation functions 11L03: Trigonometric and exponential sums, general

Keywords
representation function partition Sárközy problem

Citation

Yang, Quan-Hui; Chen, Yong-Gao. WEIGHTED REPRESENTATION FUNCTIONS ON $\mathbb{Z}_m$. Taiwanese J. Math. 17 (2013), no. 4, 1311--1319. doi:10.11650/tjm.17.2013.2463. https://projecteuclid.org/euclid.twjm/1499706119


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