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A UNIQUE REPRESENTATION BI-BASIS FOR THE INTEGERS. II
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 94 / Issue 1 / August 2016
- Published online by Cambridge University Press:
- 08 January 2016, pp. 1-6
- Print publication:
- August 2016
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- Article
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For
$n\in \mathbb{Z}$ and 
$A\subseteq \mathbb{Z}$, define 
$r_{A}(n)$ and 
${\it\delta}_{A}(n)$ by 
$r_{A}(n)=\#\{(a_{1},a_{2})\in A^{2}:n=a_{1}+a_{2},a_{1}\leq a_{2}\}$ and 
${\it\delta}_{A}(n)=\#\{(a_{1},a_{2})\in A^{2}:n=a_{1}-a_{2}\}$. We call 
$A$ a unique representation bi-basis if 
$r_{A}(n)=1$ for all 
$n\in \mathbb{Z}$ and 
${\it\delta}_{A}(n)=1$ for all 
$n\in \mathbb{Z}\setminus \{0\}$. In this paper, we prove that there exists a unique representation bi-basis 
$A$ such that 
$\limsup _{x\rightarrow \infty }A(-x,x)/\sqrt{x}\geq 1/\sqrt{2}$.
