Partitions of \(\mathbb Z_m\) with the same weighted representation functions
type: Item
instance of: scholarly article
title: Partitions of \(\mathbb Z_m\) with the same weighted representation functions
zbMATH Open document ID: 1305.11011
author: Zhenhua Qu
published in: The Electronic Journal of Combinatorics
publication date: 2014-09-04T00:00:00Z
full work available at URL: v21i2p55
review text: Summary: Let \(\mathbf{k}=(k_1,k_2,\cdots,k_t)\) be a \(t\)-tuple of integers, and \(m\) be a positive integer. For a subset \(A\subset\mathbb Z_m\) and any \(n\in\mathbb Z_m\), let \(r_A^{\mathbf{k}}(n)\) denote the number of solutions of the equation \(k_1a_1+\cdots+k_ta_t=n\) with \(a_1,\cdots,a_t\in A\). In this paper, we give a necessary and sufficient condition on \((\mathbf{k},m)\) such that there exists a subset \(A\subset \mathbb Z_m\) satisifying \(r_{A}^{\mathbf{k}}=r_{\mathbb Z_m\backslash A}^{\mathbf{k}}\). This settles a problem of \textit{Q.-H. Yang} and \textit{Y.-G. Chen} [Taiwanese J. Math. 17, No. 4, 1311--1319 (2013; Zbl 1283.11024)].
Mathematics Subject Classification ID: 11B34
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 05A17
zbMATH DE Number: 6340222
zbMATH Keywords: representation function
zbMATH Keywords: partition
zbMATH Keywords: Sárközy problem
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6340222