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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The Markoff spectrum and minima of indefinite binary quadratic forms
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by Mary E. Gbur PDF
Proc. Amer. Math. Soc. 63 (1977), 17-22 Request permission

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
  • T. W. Cusick, The connection between the Lagrange and Markoff spectra, Duke Math. J. 42 (1975), no. 3, 507–517. MR 374040
  • 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
    • L. E. Dickson, Introduction to the theory of numbers, Univ. of Chicago Press, Chicago, 1929, Chap. VII.
  • G. A. Freĭman, Non-coincidence of the spectra of Markov and of Lagrange, Mat. Zametki 3 (1968), 195–200 (Russian). MR 227110
  • Marshall Hall Jr., The Markoff spectrum, Acta Arith. 18 (1971), 387–399. MR 296023, DOI 10.4064/aa-18-1-387-399
  • V. I. Bityuckov, E. B. Dynkin, and G. E. Šilov (eds.), Matematika v SSSR za sorok let:1917–1957, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1959 (Russian). Vol. I: Survey articles; Vol. II: Biobibliography. MR 0115874
  • Ȧke Lindgren, One-sided minima of indefinite binary quadratic forms and one-sided Diophantine approximations, Ark. Mat. 13 (1975), no. 2, 287–302. MR 387197, DOI 10.1007/BF02386210
    • A. Markoff, Sur les formes binaires indéfinies, Math. Ann. 15 (1879), 381-406. O. Perron, Die Lehre von der Kettenbrüche, Chelsea, New York, 1929.
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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 63 (1977), 17-22
  • MSC: Primary 10E20; Secondary 10F20
  • DOI: https://doi.org/10.1090/S0002-9939-1977-0434963-2
  • MathSciNet review: 0434963