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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

The Markoff spectrum and minima of indefinite binary quadratic forms


Author: Mary E. Gbur
Journal: Proc. Amer. Math. Soc. 63 (1977), 17-22
MSC: Primary 10E20; Secondary 10F20
MathSciNet review: 0434963
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Abstract: Recently T.W. Cusick has shown that the Lagrange spectrum is the closure of the Markoff values of completely periodic doubly infinite sequences. In this note it is proved that if the Markoff value of a sequence M is attained at least three times, then M is a completely periodic sequence. We also show that if the minimum of an indefinite binary quadratic form is attained at least three times, then the form is equivalent to a rational form.


References [Enhancements On Off] (What's this?)

  • [1] T. W. Cusick, The connection between the Lagrange and Markoff spectra, Duke Math. J. 42 (1975), no. 3, 507–517. MR 0374040 (51 #10240)
  • [2] B. N. Delone and D. K. Faddeev, Theory of Irrationalities of Third Degree, Acad. Sci. URSS. Trav. Inst. Math. Stekloff, 11 (1940), 340 (Russian). MR 0004269 (2,349d)
  • [3] L. E. Dickson, Introduction to the theory of numbers, Univ. of Chicago Press, Chicago, 1929, Chap. VII.
  • [4] G. A. Freĭman, Non-coincidence of the spectra of Markov and of Lagrange, Mat. Zametki 3 (1968), 195–200 (Russian). MR 0227110 (37 #2695)
  • [5] Marshall Hall Jr., The Markoff spectrum, Acta Arith. 18 (1971), 387–399. MR 0296023 (45 #5084)
  • [6] Matematika v SSSR za sorok let:1917–1957, Vol. I: Survey articles. Vol. II: Biobibliography, Edited by A. G. Kuroš (ed.-in-chief), V. I. Bityuckov, V. G. Boltyanskiĭ, E. B. Dynkin, G. E. Šilov, A. P. Yuškevič. Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1959. Vol. I, 1959 (Russian). MR 0115874 (22 #6672)
  • [7] Ȧke Lindgren, One-sided minima of indefinite binary quadratic forms and one-sided Diophantine approximations, Ark. Mat. 13 (1975), no. 2, 287–302. MR 0387197 (52 #8042)
  • [8] A. Markoff, Sur les formes binaires indéfinies, Math. Ann. 15 (1879), 381-406.
  • [9] O. Perron, Die Lehre von der Kettenbrüche, Chelsea, New York, 1929.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0434963-2
PII: S 0002-9939(1977)0434963-2
Article copyright: © Copyright 1977 American Mathematical Society