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ON A PROBLEM OF CHEN AND LEV

Part of: Sequences and sets

Published online by Cambridge University Press:  28 November 2018

SHI-QIANG CHEN ,
MIN TANG  and
QUAN-HUI YANG
SHI-QIANG CHEN
Affiliation:
School of Mathematics and Statistics, Anhui Normal University, Wuhu 241003, PR China email csq20180327@163.com
MIN TANG*
Affiliation:
School of Mathematics and Statistics, Anhui Normal University, Wuhu 241003, PR China email tmzzz2000@163.com
QUAN-HUI YANG
Affiliation:
School of Mathematics and Statistics, Nanjing University of Information, Science and Technology, Nanjing 210044, PR China email yangquanhui01@163.com
*
✉tmzzz2000@163.com
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Abstract

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For a given set $S\subset \mathbb{N}$, $R_{S}(n)$ is the number of solutions of the equation $n=s+s^{\prime },s<s^{\prime },s,s^{\prime }\in S$. Suppose that $m$ and $r$ are integers with $m>r\geq 0$ and that $A$ and $B$ are sets with $A\cup B=\mathbb{N}$ and $A\cap B=\{r+mk:k\in \mathbb{N}\}$. We prove that if $R_{A}(n)=R_{B}(n)$ for all positive integers $n$, then there exists an integer $l\geq 1$ such that $r=2^{2l}-1$ and $m=2^{2l+1}-1$. This solves a problem of Chen and Lev [‘Integer sets with identical representation functions’, Integers16 (2016), A36] under the condition $m>r$.

Keywords

partitionrepresentation functioncharacteristic function

MSC classification

Primary: 11B34: Representation functions
Secondary: 11B13: Additive bases, including sumsets
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 99 , Issue 1 , February 2019 , pp. 15 - 22
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

The second author was supported by the National Natural Science Foundation of China, Grant No. 11471017. The third author was supported by the National Natural Science Foundation for Youth of China, Grant No. 11501299, the Natural Science Foundation of Jiangsu Province, Grant Nos. BK20150889 and 15KJB110014, and the Startup Foundation for Introducing Talent of NUIST, Grant No. 2014r029.

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