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UNIQUE REPRESENTATION BI-BASIS FOR THE INTEGERS

Part of: Sequences and sets

Published online by Cambridge University Press:  12 September 2013

RAN XIONG  and
MIN TANG
RAN XIONG
Affiliation:
School of Mathematics and Computer Science, Anhui Normal University, Wuhu 241003, China email ranxiong2012@163.com
MIN TANG*
Affiliation:
School of Mathematics and Computer Science, Anhui Normal University, Wuhu 241003, China email ranxiong2012@163.com
*
✉tmzzz2000@163.com
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Abstract

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For $n\in \mathbb{Z} $ and $A\subseteq \mathbb{Z} $, let ${r}_{A} (n)= \# \{ ({a}_{1} , {a}_{2} )\in {A}^{2} : n= {a}_{1} + {a}_{2} , {a}_{1} \leq {a}_{2} \} $ and ${\delta }_{A} (n)= \# \{ ({a}_{1} , {a}_{2} )\in {A}^{2} : n= {a}_{1} - {a}_{2} \} $. We call $A$ a unique representation bi-basis if ${r}_{A} (n)= 1$ for all $n\in \mathbb{Z} $ and ${\delta }_{A} (n)= 1$ for all $n\in \mathbb{Z} \setminus \{ 0\} $. In this paper, we construct a unique representation bi-basis of $ \mathbb{Z} $ whose growth is logarithmic.

Keywords

bi-basisrepresentation function

MSC classification

Secondary: 11B34: Representation functions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 89 , Issue 3 , June 2014 , pp. 460 - 465
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

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