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On a generalization of Markoff’s theory. (Sur une généralisation de la théorie de Markoff.) (French) Zbl 0714.11039

The author generalizes the phenomenon of Markoff forms using the equation \[ (-\epsilon_ 1A)m^ 2+\epsilon_ 2m^ 2_ 1+\epsilon_ 1m^ 2_ 2=(a+r+1)mm_ 1m_ 2 \] for \(\epsilon =\pm 1\). Using results of R. Remak [Math. Ann. 92, 155–182 (1924; JFM 50.0099.02)] and J. W. S. Cassels [An introduction to diophantine approximation, Cambridge Tract 45 (1957; Zbl 0077.04801)], he constructs trees of forms for which the Markoff spectrum accumulates at \(1/N\) (for integers \(N>3\)).
Reviewer: Harvey Cohn

MSC:

11J06 Markov and Lagrange spectra and generalizations
11D09 Quadratic and bilinear Diophantine equations
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References:

[1] Markoff, A. A., Math. Ann., 17, 379-399 (1880) · JFM 12.0143.02
[2] Cassels, J. W.S., An introduction to diophantine approximation, (Cambridge Tracts in Mathematics, Vol. 45 (1957), Cambridge Univ. Press: Cambridge Univ. Press London/New York) · Zbl 0077.04801
[3] Remak, R., Uber indefinite binäre quadratische minimal Formen, Math. Ann., 92, 155-182 (1924) · JFM 50.0099.02
[4] Frobenius, G., Uber die Markoffschen Zahlen, Preussiche Akad. Wiss. S.B., 458-487 (1913) · JFM 44.0255.01
[5] Perron, O., Uber die Approximation irrationaler Zahlen durch rationale, I, (Sitzungsber. Heidelb. Akad. Wiss. (1921), Springer-Verlag: Springer-Verlag New York/Berlin), Abh. 8 · JFM 48.0193.01
[6] Cusick, T. W.; Flahive, M. E., The Markoff and Lagrange spectra, (Mathematical Surveys and Monographs, Vol. 30 (1989), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI) · Zbl 0685.10023
[7] Perrine, S., Approximation diophantienne, Thèse présentée à l’Université de Metz (Décember 1988), (Théorie de Markoff)
[8] Dickson, L. E., (History of a Theory of Numbers (1952), Chelsea: Chelsea New York)
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