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UNIQUE REPRESENTATION BI-BASIS FOR THE INTEGERS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 89 / Issue 3 / June 2014
- Published online by Cambridge University Press:
- 12 September 2013, pp. 460-465
- Print publication:
- June 2014
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- Article
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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.
