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


11J06 Markov and Lagrange spectra and generalizations
11D09 Quadratic and bilinear Diophantine equations
Full Text: DOI


[1] Markoff, A.A.; Markoff, A.A., Sur LES formes quadratiques binaires indéfinies, Math. ann., Math. ann., 17, 379-399, (1880) · JFM 12.0143.02
[2] Cassels, J.W.S., An introduction to Diophantine approximation, () · 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.; Perron, O., Uber die approximation irrationaler zahlen durch rationale, I, (), Abh. 8 · JFM 48.0193.01
[6] Cusick, T.W.; Flahive, M.E., The markoff and Lagrange spectra, () · Zbl 0685.10023
[7] Perrine, S., Approximation diophantienne, Thèse présentée à l’université de Metz, (Décember 1988), (Théorie de Markoff)
[8] Dickson, L.E., ()
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.