Vahlen theorem for two-dimensional convergents
Given \(\underline{\alpha}= (\alpha_1,\dots, \alpha_d)\in \mathbb{R}^d\), \(q\in \mathbb{N}\), let \(a_j\) be the nearest integers to the numbers \(qa_j\). A vector \((a_1,\dots, a_d;q)\) is called convergent of first type to \(\underline{\alpha}\) if the system of inequalities \(\|q'\alpha_j\|\leq \|q\alpha_j\|\) \((j= 1,\dots, d)\) is satisfied for no other positive integer \(q'< q\). Let \(\Delta(\underline{\alpha})\) denote the set of all ``denominators'' \(q\) that determine convergents uniquely. This set is infinite except when \(\underline{\alpha}\) is rational. Further, given \(\underline{\alpha}\in \mathbb{R}^d\), \({\mathbf q}= (q_1,\dots, q_d)\in \mathbb{Z}^d\), \({\mathbf q}\neq 0\), let \(a\) be the integer nearest to \({\mathbf {qa}}\). A vector \((a; q_1,\dots, q_d)\) is called convergent of second type to \(\alpha\) if for any other integer vector \({\mathbf q}'\neq 0\) with \(|q_j'|\leq |q_j|\) the following inequalities are satisfied: \[ \begin{aligned} \|{\mathbf {qa}}\|&< \|{\mathbf q}' {\mathbf a}\|\quad \text{for} \quad ((|q_1'|,\dots, |q_d'|)\neq (|q_1|,\dots, |q_d|), \qquad \text{and}\\ \|{\mathbf {qa}}\|&\leq \|{\mathbf q}' {\mathbf a}\|\quad \text{for} \quad ((|q_1'|,\dots, |q_d'|)= (|q_1|,\dots, |q_d|). \end{aligned} \] Let \(\Delta^* (\underline{\alpha})\) denote the set of all ``denominators'' \({\mathbf q}\) of convergents of the second type. The author proves the following results which both generalize Vahlen's theorem for ordinary continued fractions. (I) Given \(\underline{\alpha}= (\alpha_1, \alpha_2)\) and \(q\in \Delta(\underline{\alpha})\), let \(q'\in \Delta(\underline{\alpha})\) be the least denominator such that \(q< q'\), \(\|q'\alpha_1\|\leq \|q\alpha_1\|\), \(\|q'\alpha_2\|\leq \|q\alpha_2\|\). Then \[ \min \{q\|q\alpha_1\|\|q\alpha_2\|,\;q'\|q'\alpha_1\|\|q'\alpha_2\|\}< \tfrac{1}{2}. \] (II) Given \(\underline{\alpha}= (\alpha_1, \alpha_2)\) and \({\mathbf q}\in \Delta^*(\underline{\alpha})\), let \({\mathbf q}'\in \Delta^*(\underline{\alpha})\) be the denominator with \(q_1'< q_1\), \(q_2'< q_2\), \(\|q_1\alpha_1+ q_2\alpha_2\|< \|q_1'\alpha_1+ q_2'\alpha_2\|\) that yields the smallest value of \(\|q_1'\alpha_1+ q_2'\alpha_2\|\). Then \[ \min\{|q_1q_2|\|q_1\alpha_1+ q_2\alpha_2\|,\;|q_1'q_2'|\|q_1'\alpha_1+ q_2'\alpha_2\|\}< \tfrac{1}{2}. \] The proof is based on Voronoi's concept of minimal nodes in a three-dimensional lattice.
scientific article; zbMATH DE number 1429646