UNIQUE REPRESENTATION BI-BASIS FOR THE INTEGERS
Ran Xiong, Min Tang · Bulletin of the Australian Mathematical Society · 2013
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.