A UNIQUE REPRESENTATION BI-BASIS FOR THE INTEGERS. II

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 Aa unique representation bi-basis if r_{A}(n)=1for all n\in \mathbb{Z}and {\it\delta}_{A}(n)=1for all n\in \mathbb{Z}\setminus \{0\}. In this paper, we prove that there exists a unique...