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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[1] Cassels, J.W.S., An introduction to diophantine approximation. Cambridge Tracts in Mathematics and Mathematical Physics, No. 45, Cambridge University Press, New-York, 1957. · Zbl 0077.04801
[2] Colliot-Thélène, J.L., Les grands thèmes de François Châtelet. Enseig. Math.(2) 34 (1988), no. 3-4, 387-405. · Zbl 0678.01008
[3] Cusick, T.W., On Perrine’s generalized Markoff equation. Aequationes Mathematicae46 (1993), 203-211. · Zbl 0813.11013
[4] Cusick, T.W. and Flahive, M.E., The Markoff and Lagrange spectra. Mathematical surveys and monographs, 30. American Mathematical Society, Providence, RI, 1989. · Zbl 0685.10023
[5] Markoff, A.A., Sur les formes quadratiques binaires indéfinies. Math. Ann.6 (1879), 381-406; Math Ann.17 (1880), 379-399.
[6] Perrine, S., Approximation diophantienne (Théorie de Markoff). Thèse présentée à l’Université de Metz (décembre 1988).
[7] Perrine, S., Sur une généralisation de la théorie de Markoff. J. Number Theory37 (1991), 211-230. · Zbl 0714.11039
[8] Perrine, S., Sur des équations diophantiennes généralisant celle de Markoff. Ann. Fac. Sci. Toulouse Math.(6) 6 (1997), no. 1, 127-141. · Zbl 0882.11020
[9] Schmidt, A.L., Minimum et quadratic forms with respect to fuchsian groups (I). J. Reine Angew. Math.286/287 (1976), 341-348. · Zbl 0332.10015
[10] Segre, B., Arithmetic upon an algebraic surface. Bull. Amer. Math. Soc.51 (1945), 152-161. · Zbl 0061.07105
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