Diophantine aspects of the Calkin-Wilf iteration
The Calkin-Wilf tree is defined as a binary tree with a root \(1/1\) and the left and right child of a node \(a/b\) are \(a/(a+b)\) and \((a+b)/b\), respectively. In this paper, the authors studied new diophantine aspects of this tree as follows. Let \(U(z)=z+1\) and \(T(z)=z+1\) be two Möbius transformations. For given \(\alpha\) a complex number, the \(\alpha\)-Calkin-Wilf tree is defined as a binary tree with a root \(\alpha\) and the left and right child of a node \(z\) are \(U(z)\) and \(T(z)\), respectively. The authors described the relation of the \(\alpha\)-Calkin-Wilf iteration to continued fractions. Then they showed that the \(\alpha\)-Calkin-Wilf tree contains any positive number \(\beta\) of the form \(\beta=\frac{a\alpha+b}{c\alpha+d}\) with \(a,b,c,d\geq1\) and \(ad-bc=1\). all these numbers \(\beta\) have the same tail in their continued fraction expansion as \(\alpha\).
scientific article; zbMATH DE number 5919429