Unique representation bi-basis for the integers

R. Xiong; M. Tang

Unique representation bi-basis for the integers

Authors

R. Xiong
M. Tang

Keywords:

bi-basis, representation function

Abstract

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. DOI: 10.1017/S0004972713000762

Published

2014-03-25

Issue

Vol. 89 No. 3 (2014)

Section

Articles

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