Unique representation bi-basis for rational numbers field

Abstract

For $A\subseteq {\mathbb {Q}}$, $\alpha \in {\mathbb {Q}}$, let $r_{A}(\alpha )=\#\{(a_{1}, a_{2})\in A^{2}: \alpha =a_{1}+a_{2}, a_{1}\le a_{2}\},$ $\delta _{A}(\alpha )=\#\{(a_{1}, a_{2})\in A^{2}: \alpha =a_{1}-a_{2} \}.$ In this paper, we construct a set $A\subset {\mathbb {Q}}$ such that $r_{A}(\alpha )=1$ for all $\alpha \in {\mathbb {Q}}$ and $\delta _{A}(\alpha )=1$ for all $\alpha \in {\mathbb {Q}}\setminus \{{0}\}$.