A tree of approximation constants analogous to those of Markov’s diophantine equation. (Un arbre de constantes d’approximation analogue à celui de l’équation diophantienne de Markoff.) (French) Zbl 0924.11057

To each triplet of positive integers solving \(m^2 + m_{1}^2 + m_{2}^2 = 4 m m_1 m_2 - m\), the author naturally associates a quadratic form over the rationals. For each \(\theta\), the usual related constant of diophantine approximation is \(C(\theta) = \lim \inf_q | | q \theta| | \). A main result of the paper is that if \(\theta\) is a root (in the standard manner) of one of the aforementioned forms, then \(C(\theta) = m/\sqrt{16 m^2 - 4}\). Furthermore, there exist uncountably many irrational \(\alpha\) such that \(C(\alpha) = 4\). This paper builds upon previous work of the author, especially [Ann. Fac. Sci. Toulouse Math. 6, 127-141 (1997; Zbl 0882.11020)] and T. Cusick’s [Aequ. Math. 46, 203-211 (1993; Zbl 0813.11013)] completion of the author’s earlier treatments of solutions of this and related generalized Markoff equations.


11J06 Markov and Lagrange spectra and generalizations
11D25 Cubic and quartic Diophantine equations
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