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Markov and Lagrange spectra (survey of the literature)

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A survey is given of investigations on the Markov problem of the arithmetic minima of indeterminate, binary, quadratic forms and on the Lagrange-Hurwitz problem of Diophantine approximations of irrational numbers by rational numbers.

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Literature cited

  1. P. Bachmann, Die Arithmetik der quadratischen Formen, 2. Abt. (Zahlentheorie, IV2), Leipzig-Berlin (1923).

  2. I. Borosh, “Numerical evidence on the uniqueness of Markoff numbers,” Notices Am. Math. Soc.,21, A-55, Abstract No. 711-10-32 (1974).

    Google Scholar 

  3. I. Borosh, “More numerical evidence on the uniqueness of Markov numbers,” BIT,15, No. 4, 351–357 (1975).

    Google Scholar 

  4. R. T. Bumby, “The Markov spectrum,” in: Diophantine Approximation and Its Applications, New York (1973), pp. 25–49.

  5. R. T. Bumby, “Structure of the Markoff spectrum below √12,” Acta Arithm.,29, 299–307 (1976).

    Google Scholar 

  6. J. W. S. Cassels, “The Markoff chain,” Ann. Math. (2),50, 676–685 (1949).

    Google Scholar 

  7. J. W. S. Cassels, “Uber einen Perronschen Satz,” Arch. Math.,3, 10–14 (1952).

    Google Scholar 

  8. J. W. S. Cassels, Introduction to Diophantine Approximation, CUP, London (1957).

    Google Scholar 

  9. H. Conn, “Approach to Markoff's minimal forms through modular functions,” Proc. Int. Congr. Math., Amsterdam (1954), Vol. 2, pp. 10–11.

    Google Scholar 

  10. H. Cohn, “Approach to Markoff's minimal forms through modular functions,” Ann. Math. (2),61, No. 1, 1–12 (1955).

    Google Scholar 

  11. H. Cohn, “Representations of Markoff's binary quadratic forms by geodesics on a perforated torus,” Acta Arithm.,18, 125–136 (1971).

    Google Scholar 

  12. H. Cohn, “Markoff forms and primitive words,” Math. Ann.,196, No. 1, 8–22 (1972).

    Google Scholar 

  13. H. Cohn, “Some direct limits of primitive homotopy words and of Markoff geodesics,” in: Discontinuous Groups and Riemann Surfaces, Ann. Math. Studies, Vol. 79 (1974), pp. 81–98.

    Google Scholar 

  14. T. W. Cusick, “Sums and products of continued fractions,” Proc. Am. Math. Soc.,27, No. 1, 35–38 (1971).

    Google Scholar 

  15. T. W. Cusick, “On M. Hall's continued fraction theorem,” Proc. Am. Math. Soc.,38, No. 2, 253–254 (1973).

    Google Scholar 

  16. T. W. Cusick, “The largest gaps in the lower Markoff spectrum,” Duke Math. J.,41, No. 2, 453–463 (1974).

    Google Scholar 

  17. T. W. Cusick, “The connection between the Lagrange and Markoff spectra,” Duke Math. J.,42, No. 3, 507–517 (1975).

    Google Scholar 

  18. T. W. Cusick and R. A. Lee, “Sums of sets of continued fractions,” Proc. Am. Math. Soc.,30, No. 2, 241–246 (1971).

    Google Scholar 

  19. H. Davenport and H. Heilbronn, “On the minimum of a bilinear form,” Q. J. Math.,18, No. 70, 107–121 (1947).

    Google Scholar 

  20. C. S. Davis, “The minimum of an indefinite binary quadratic form,” Q. J. Math. (2),1, 241–242 (1950).

    Google Scholar 

  21. N. Davis and J. R. Kinney, “Quadratic irrationals in the lower Lagrange spectrum,” Can. J. Math.,25, No. 3, 578–584 (1973).

    Google Scholar 

  22. L. E. Dickson, Studies in the Theory of Numbers, Chicago (1930).

  23. B. Divis, “On the sums of continued fractions,” Acta Arithm.,22, No. 2, 157–173 (1973).

    Google Scholar 

  24. L. R. Ford, “On the closeness of approach of complex rational fractions to a complex irrational number,” Trans. Am. Math. Soc.,27, 146–154 (1925).

    Google Scholar 

  25. G. Frobenius, “Uber die Markoffschen Zahlen,” Sitzungsber. Preuss. Akad. Wiss., 458–487 (1913).

  26. M. Fujiwara, “Zahlengeometrische Untersuchung uber die extremen Formen fur die indefiniten quadratischen Formen,” Math. Ann.,85, 21–25 (1922).

    Google Scholar 

  27. M. Fujiwara, “Zur Theorie der binaren indefiniten quadratischen Formen,” Tohoku Math. J.,23, 76–89 (1924).

    Google Scholar 

  28. M. E. Gbur, “On the lower Markov spectrum,” Monatsh. Math.,81, No. 2, 95–107 (1976).

    Google Scholar 

  29. M. Hall, Jr., “On the sum and product of continued fractions,” Ann. Math. (2),48, 966–993 (1947).

    Google Scholar 

  30. M. Hall, Jr., “The Markoff spectrum,” Acta Arithm.,18, 387–399 (1971).

    Google Scholar 

  31. C. J. Hightower, “On the minima of real indefinite binary quadratic forms,” Diss., Tulane Univ. (1963).

  32. C. J. Hightower, “The minima of indefinite binary quadratic forms,” J. Number Theory,2, No. 3, 364–378 (1970).

    Google Scholar 

  33. N. Hofreiter, “Diophantische Approximationen in imaginaren quadratischen Zahlkorpern,” Monatsh. Math. Phys.,45, 175–190 (1936).

    Google Scholar 

  34. G. Humbert, “Sur les fractions continues et les formes quadratiques binaires indéfinies,” C. R. Acad. Sci. Paris,162, 23–26 (1916).

    Google Scholar 

  35. G. Humbert, “Sur les fractions continues ordinaires et les formes quadratiques binaires indéfinies,” J. Math. Pure Appl. (7),2, 104–154 (1916).

    Google Scholar 

  36. A. Hurwitz, “Uber die angenaherte Darstellung der Irrationalzahlen durch rationale Bruche. I, II,” Math. Ann.,39, 279–284 (1891);44, 417–426 (1894).

    Google Scholar 

  37. A. Hurwitz, “Uber eine Aufgabe der unbestimmten Analysis,” Arch. Math. Phys. (3),11, 185–196 (1906).

    Google Scholar 

  38. T. H. Jackson, “Note on the minimum of an indefinite binary quadratic form,” J. London Math. Soc. (2),5, 209–214 (1972).

    Google Scholar 

  39. J. R. Kinney and T. S. Pitcher, “The Hausdorff-Besicovich dimension of level sets of Perron's modular functions,” Trans. Am. Math. Soc.,124, 122–130 (1966).

    Google Scholar 

  40. J. R. Kinney and T. S. Pitcher, “On the lower range of Perron's modular function,” Can. J. Math.,21, No. 4, 808–816 (1969).

    Google Scholar 

  41. J. F. Koksma, Diophantische Approximationen, Berlin (1936).

  42. C. G. Lekkerkerker, “Geometric deduction of Markov's minimal forms,” Math. Centrum, Amsterdam, Report zw 1959-008.

  43. C. G. Lekkerkerker, Geometry of Numbers, Amsterdam (1969).

  44. A. Lindgren, “Asymmetric and one-sided minima of indefinite binary quadratic forms,” Uppsala Univ., Dept. of Math., Report No. 6 (1975).

  45. A. Lindgren, “One-sided minima of indefinite binary quadratic forms and one-sided diophantine approximations,” Arkiv Math.,13, No. 2, 287–302 (1975).

    Google Scholar 

  46. O. Perron, “Uber die Approximation irrationaler Zahlen durch rationale. I, II,” Sitzungsberichte, Heidelberg. Akad. Wiss.,4, 17 (1921);8, 12 (1921).

    Google Scholar 

  47. O. Perron, “Uber die Approximation einer komplexen Zahl durch Zahlen des Korpers K(i). I, II,” Math. Ann.,103, 533–544 (1930);105, 160–164 (1931).

    Google Scholar 

  48. O. Perron, “Uber einen Approximationssatz von Hurwitz und uber die Approximation einer komplexen Zahl durch Zahlen des Körpers der dritten Einheitswurzeln,” Sitzungsberichte, Bayer. Akad. Wiss., Math.-Natur. Abteilung, 129–154(1931).

  49. O. Perron, “Diophantische Approximationen in imaginaren quadratischen Zahlkorpern insbesondere im Körper R.\(R(i\sqrt 2 )\),” Math. Z.,37, 749–767 (1933).

    Google Scholar 

  50. G. Poitou, “Sur l'approximation des nombres complexes par les nombres des corps imaginaires quadratiques dénués d'idéaux non principaux particulierement lorsque vaut l'algorithme d'Euclide,” Ann. Scient. Ecol. Norm. Super. Paris (3),70, No. 3, 199–265 (1953).

    Google Scholar 

  51. R. A. Rankin, “Diophantine approximation and horocyclic groups,” Can. J. Math.,9, No. 2, 277–290 (1957).

    Google Scholar 

  52. R. Remak, “Uber indefinite binare quadratische Minimalformen,” Math. Ann.,92, 155–182 (1924).

    Google Scholar 

  53. R. Remak, “Uber die geometrische Darstellung der indefiniten binaren quadratischen Minimalformen,” Jber. Deutsch. Math. Vereining.,33, 228–245 (1925).

    Google Scholar 

  54. D. Rosen and G. S. Patterson, Jr., “Some numerical evidence concerning the uniqueness of the Markov numbers,” Math. Comput.,25, No. 116, 919–921 (1971).

    Google Scholar 

  55. G. Rosenberger, “Fuchssche Gruppen, die freies Produktzyklischer Gruppen sind, und die Gleichung x2+y2+z2=xyz,” Math. Ann.,199, 213–227.

  56. G. Rosenberger, “The uniqueness of the Markoff numbers,” Math. Comput.,30, No. 134, 361–365 (1976).

    Google Scholar 

  57. H. Schecker, “Uber die Menge der Zahlen, die als Minima binarer quadratischer Formen auftreten,” Diss. Univ. Dortmund (1972).

  58. A. L. Schmidt, “Farey triangles and Farey quadrangles in the complex plane,” Math. Scand.,21, 241–295 (1967).

    Google Scholar 

  59. A. L. Schmidt, “Farey simplices in the space of quaternions,” Math. Scand.,24, 31–65 (1969).

    Google Scholar 

  60. A. L. Schmidt, “On the approximation of quaternions,” Math. Scand.,34, 184–186 (1974).

    Google Scholar 

  61. A. L. Schmidt, “Diophantine approximation of complex numbers,” Acta Math.,134, No. 1–2, 1–85 (1975).

    Google Scholar 

  62. A. L. Schmidt, “On C-minimal forms,” Math. Ann.,215, 203–214 (1975).

    Google Scholar 

  63. A. L. Schmidt, “Minimum of quadratic forms with respect to Fuchsian groups. I, II.”

  64. I. Schur, “Zur Theorie der indefiniten binaren quadratischen Formen,” Sitzungsber. Preuss. Akad. Wiss., 212–213 (1913).

  65. P. Varnavides, “Antisymmetric Markoff forms,” Indag. Math.,20, 463–469 (1958).

    Google Scholar 

  66. M. Voicu, “Studiul ecuatiei ℳ (α) =n pentru n⩾3 O generalizare a teoreniei lui Perron,” Stud. Si Cerc. Mat.,22, No. 7, 1101–1108 (1970).

    Google Scholar 

  67. M. Voicu, “Studiul functiei lui Perron asupra inecuatiei\(3< \mathcal{M}(\alpha )< \sqrt {12} \),” Stud. Si Cerc. Mat.,23, No. 10, 1587–1592 (1971).

    Google Scholar 

  68. A. A. Bershtein, “On necessary and sufficient conditions that points of the Markov spectrum be in the Lagrange spectrum,” Dokl. Akad. Nauk SSSR,191, No. 5, 971–973 (1970).

    Google Scholar 

  69. A. A. Bershtein, G. V. Pavlova, and G. A. Freiman, “The Markov spectrum of the arithmetic minima of indeterminate binary quadratic forms,” in: Number-Theoretic Investigations on the Markov Spectrum and the Structural Theory of the Addition of Sets [in Russian], Kal. GU, Moscow (1973), pp. 3–125.

    Google Scholar 

  70. G. A. Freiman, “Noncoincidence of the Markov and Lagrange spectra,” in: Number-Theoretic Investigations on the Markov Spectrum and the Structural Theory of the Addition of Sets [in Russian], Kal. GU, Moscow (1973), pp. 10–15.

    Google Scholar 

  71. A. A. Bershtein, “On the connection between the Markov and Lagrange spectra,” in: Number-Theoretic Investigations on the Markov Spectrum and the Structural Theory of the Addition of Sets [in Russian], Kal. GU, Moscow (1973), pp. 16–49.

    Google Scholar 

  72. A. A. Bershtein, “On the structure of the Markov spectrum,” in: Number-Theoretic Investigations on the Markov Spectrum and the Structural Theory of the Addition of Sets [in Russian], Kal. GU, Moscow (1973), pp. 50–78.

    Google Scholar 

  73. G. V. Pavlova and G. A. Freiman, “On the part of the Markov spectrum of measure zero,” in: Number-Theoretic Investigations on the Markov Spectrum and the Structural Theory of the Addition of Sets [in Russian], Kal. GU, Moscow (1973), pp. 79–86.

    Google Scholar 

  74. G. A. Freiman, “On the origin of the Hall ray,” in: Number-Theoretic Investigations on the Markov Spectrum and the Structural Theory of the Addition of Sets [in Russian], Kal. GU, Moscow (1973), pp. 87–120.

    Google Scholar 

  75. A. A. Bershtein, “On the structure and relations of the Markov and Lagrange spectra,” Author's Abstract of Candidate's Dissertation, Kal. GU, Kalinin (1974).

    Google Scholar 

  76. B. A. Venkov, “The metric of Lobachevskii and the metric of Voronyi in the geometry of numbers,” Proc. 3rd All-Union Math. Congress,3, 14–21 (1958).

    Google Scholar 

  77. A. Vinogradov, B. Delone, and D. Fuks, “On rational approximations to irrational numbers with bounded incomplete quotients,” Dokl. Akad. Nauk SSSR,118, No. 5, 862–865 (1958).

    Google Scholar 

  78. L. Ya. Vulakh, “On the Markov spectrum of imaginary quadratic fields\(\mathbb{Q}(i\sqrt \mathcal{D} ),where\mathcal{D}3\),” Vestn. Mosk. Univ., No. 6, 32–41 (1971).

    Google Scholar 

  79. L. Ya. Vulakh, “On the Markov spectrum of the Gauss field,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 5, 27 (1971).

    Google Scholar 

  80. L. Ya. Vulakh, “On the Markov spectrum of imaginary quadratic fields,” Author's Abstract of Candidate's Dissertation, Moscow State Univ. (1971).

  81. L. Ya. Vulakh, “Diophantine closed sets,” Tr. Mosk. Inst. Radiotekh., Elektron. Avtom., No. 52, 27–31 (1971).

    Google Scholar 

  82. L. Ya. Vulakh, “The Diophantine equation p 21 + p 22 + 5q2 = 5p1p2q and the Markov spectrum,” Tr. Mosk. Inst. Radiotekh., Elektron. Avtom., No. 57, 54–58 (1972).

    Google Scholar 

  83. L. Ya. Vulakh, “On the Markov spectrum of the Gauss field,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 2 (129), 26–40 (1973).

    Google Scholar 

  84. L. Ya. Vulakh, “The Diophantine equation p2 + 2q2 + 3z2=6pq2 and the Markov spectrum,” Tr. Mosk. Inst. Radiotekh., Elektron. Avtom., No. 67, 105–112 (1973).

    Google Scholar 

  85. L. Ya. Vulakh, “On an accumulation point of the Markov spectrum of the field\(\mathbb{Q}(i\sqrt \mathcal{D} )\),” Vestn. Mosk. Univ., No. 1, 9–11 (1975).

    Google Scholar 

  86. D. S. Gorshkov, “The geometry of Lobachevskii in connection with certain questions of arithmetic,” Author's Abstract of Candidate's Dissertation, Leningrad State Univ. (1953).

  87. D. S. Gorshkov, “The geometry of Lobachevskii in connection with certain questions of arithmetic,” Author's Abstract of Candidate's Dissertation, Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,67 (1977).

  88. B. N. Delone, The Petersburg School of Number Theory [in Russian], Moscow-Leningrad (1947).

  89. B. N. Delone, “On the work of A. A. Markov ‘On binary quadratic forms of positive determinant’,” Usp. Mat. Nauk,3, No. 5, 3–5 (1948).

    Google Scholar 

  90. B. N. Delone and A. M. Vinogradov, “Uber den Zusammenhang zwischen den Lagrangeschen Klassen der Irrationalitaten mit begrenzten Teilnennern und den Markoffschen Klassen der extremen Formen,” in: Leonard Euler zum 250 Geburstag, Berlin (1959), pp. 100–108.

  91. P. G. Kogoniya, “On the set of Markov numbers,” Dokl. Akad. Nauk SSSR,78, No. 4, 637–640 (1951).

    Google Scholar 

  92. P. G. Kogoniya, “On the structure of the set of Markov numbers,” Tr. Tbilissk. Mat. Inst.,19, 121–133 (1953).

    Google Scholar 

  93. P. G. Kogoniya, “On the set of ‘generalized’ Markov numbers. I, II,” Tr. Tbilissk. Univ.,56, 105–120 (1955);84, 143–149 (1961–1962).

    Google Scholar 

  94. P. G. Kogoniya, “On cluster points of the set of Markov numbers,” Dokl. Akad. Nauk SSSR,118, No. 4, 632–635 (1958).

    Google Scholar 

  95. P. G. Kogoniya, “On the set of cluster points of the set of Markov numbers,” Proc. 3rd All-Union Math, Congress (Moscow, 1956), Moscow (1956), Vol. 1, p. 5.

    Google Scholar 

  96. P. G. Kogoniya, “On the set of cluster points of the set of Markov numbers,” Tr. Tbilissk. Mat. Inst.,26, 3–16 (1959).

    Google Scholar 

  97. P. G. Kogoniya, “On the maximal cluster point of a subset of the set of Markov numbers,” Tr. Tbilissk. Mat. Inst.,26, 17–22 (1959).

    Google Scholar 

  98. P. G. Kogoniya, “On the connection between the Lagrange and Markov spectra. I–IV,” Tr. Tbilissk. Univ.,76, 161–171 (1959);102, 95–104, 105–113 (1964); Tr. Tbilissk. Mat. Inst.,29, 15–35 (1963) (Part I was called: “On the connection between the set of Markov numbers and the Markov spectrum (I)”.)

    Google Scholar 

  99. P. G. Kogoniya, “Some questions of rational approximation,” Lit. Mat. Sb.,6, No. 1, 126–128 (1966).

    Google Scholar 

  100. P. G. Kogoniya, “Some questions of rational approximation. I, II,” Tr. Tbilissk. Univ.,117, 45–62 (1966);129, 267–274 (1968).

    Google Scholar 

  101. P. G. Kogoniya, “Some problems of the theory of diophantine approximations,” Author's Abstract of Candidate's Dissertation, Leningrad State Univ. (1969).

  102. P. G. Kogoniya, “On generalized Lagrange spectra,” Tr. Tbilissk. Univ.,A1(137), 35–43 (1971).

    Google Scholar 

  103. P. G. Kogoniya, “On generalized Lagrange spectra,” Tr. Tbilissk. Univ.,A10(158), 3–5 (1975).

    Google Scholar 

  104. A. N. Korkin and E. I. Zolotarev, “Sur les formes quadratiques,” Math. Ann.,6, 366–389 (1873).

    Google Scholar 

  105. A. A. Markov, “Sur les formes binaires indefinies. I, II,” Math. Ann.,15, 381–409 (1879);17, 379–400 (1880).

    Google Scholar 

  106. A. A. Markov, “On binary quadratic forms of positive determinant,” St. Petersburg (1880); reprinted: Usp. Mat. Nauk,3, No. 5, 7–51 (1948).

  107. G. A. Freiman, “On the coincidence of the Markov and Lagrange spectra,” Mat. Zametki,3, No. 2, 195–200 (1968).

    Google Scholar 

  108. G. A. Freiman, Diophantine Approximations and the Geometry of Numbers (The Markov Problem) [in Russian], Kal. GU, Kalinin (1975).

    Google Scholar 

  109. G. A. Freiman and A. A. Yudin, “On the Markov spectrum,” Litov. Mat. Sb.,6, No. 3, 443–447 (1966).

    Google Scholar 

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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 67, pp. 5–38, 1977.

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Malyshev, A.V. Markov and Lagrange spectra (survey of the literature). J Math Sci 16, 767–788 (1981). https://doi.org/10.1007/BF01213889

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