## 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.

### MSC:

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

### Citations:

Zbl 0822.11020; Zbl 0813.11013; Zbl 0882.11020
Full Text:

### References:

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