UNIQUE REPRESENTATION BI-BASIS FOR THE INTEGERS

Abstract

AbstractFor$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.