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On the three-dimensional Vahlen theorem. (English. Russian original) Zbl 1369.11048
Math. Notes 95, No. 1, 136-138 (2014); translation from Mat. Zametki 95, No. 1, 154-156 (2014).
From the text: Let \(\gamma^{(1)}, \ldots, \gamma^{(s)}\) be the basis nodes of a full-rank lattice
\[ \Gamma = \left\{ m_1 \gamma^{(1)} + \cdots + m_s \gamma^{(s)}: m_1,\ldots, m_s \in \mathbb Z \right\} \subset \mathbb R^s. \]
Vahlen’s theorem concerning the approximation of numbers by convergents has the following interpretation in terms of lattices: for every Voronoi basis \(\{\gamma^{(1)}, \gamma^{(2)}\}\),
\[ \min\left\{\left|\gamma_1^{(1)}, \gamma_2^{(1)}\right|, \left|\gamma_1^{(2)}, \gamma_2^{(2)}\right|\right\} \leq \tfrac{1}{2} \det \Gamma. \]

MSC:
11H06 Lattices and convex bodies (number-theoretic aspects)
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