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Transcendental numbers and diophantine approximations


Author: Serge Lang
Journal: Bull. Amer. Math. Soc. 77 (1971), 635-677
MSC (1970): Primary 10F35, 10F40; Secondary 10F45, 33A10, 33A35, 32A20, 14L10
DOI: https://doi.org/10.1090/S0002-9904-1971-12761-1
MathSciNet review: 0289424
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  • 1. W. W. Adams, Asymptotic Diophantine approximations to e, Proc. Nat. Acad. Sci. U.S.A. 55 (1966), 28-31. MR 32 #4085. MR 186626
  • 2. W. W. Adams, Asymptotic Diophantine approximations and Hurwitz numbers, Amer. J. Math. 89 (1967), 1083-1108. MR 56 #5082. MR 222030
  • 3. W. W. Adams, Simultaneous asymptotic Diophantine approximations to a basis of a real cubic number field, J. Number Theory 1 (1969), 179-194. MR 39 #1409. MR 240055
  • 4. W. W. Adams, A lower bound in asymptotic Diophantine approximations, Duke Math. J. 35 (1968), 21-35. MR 36 #5083. MR 222031
  • 5. W. W. Adams, Simultaneous asymptotic Diophantine approximations, Mathematika 14 (1967), 173-180. MR 36 #3730. MR 220678
  • 6. W. W. Adams, Simultaneous asymptotic Diophantine approximation to a basis of a real number field, Nagoya Math. J. (to appear). MR 285490
  • 7. W. W. Adams, Transcendental numbers in the P-adic domain, Amer. J. Math. 88 (1966), 279-308. MR 33 #5564. MR 197399
  • 8. W. Adams and S. Lang, Some computations in Diophantine approximations, J. Reine Angew. Math. 220 (1965), 163-173. MR 32 #91. MR 182608
  • 9. L. Alaoglu and P. Erdös, On highly composite and similar numbers, Trans. Amer. Math. Soc. 56 (1944), 448-469. MR 6, 117. MR 11087
  • 10. J. Ax, On Schannuel's conjecture, Ann. of Math. (2) 93 (1971), 252-268 (and another paper to appear).
  • 11. A. Baker, Linear forms in the logarithms of algebraic numbers. I, II, III, Mathematika 13 (1966), 204-216; ibid. 14 (1967), 102-107, 220-228. MR 36 #3732. MR 258756
  • 12. A. Baker, An estimate for the ℘-function at an algebraic point, Amer. J. Math. (to appear). MR 281694
  • 13. A. Baker, On the quasi-periods of the Weierstrass zeta function, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1969, 145-157. MR 274394
  • 14. A. Baker, Contributions to the theory of Diophantine equations. I. On the representation of integers by binary forms, Philos. Trans. Roy. Soc. London Ser. A 263 (1967/68), 173-191; II. The Diophantine equation Y2=X3+k, ibid., 193-208. MR 37 #4005; #4006. MR 228424
  • 15. A. Baker, The Diophantine equation Y2=ax3+bx2+cx+d, J. London Math. Soc. 43 (1968), 1-9. MR 38 #111. MR 231783
  • 16. A. Baker, Imaginary quadratic fields with class number 2, Ann. of Math. (to appear). MR 299583
  • 17. A. Baker and J. Coates, Integer points on curves of genus 1, Proc. Cambridge Philos. Soc. 67 (1970), 595-602. MR 41 #1638. MR 256983
  • 18. A. Baker and H. Stark, On a fundamental inequality in number theory, Ann. of Math. (to appear).
  • 19. H. Behnke, Über die Verteilung von Irrationalitatenmod 1, Abh. Math. Sem. Univ. Hamburg 1 (1922), 252-267.
  • 20. H. Behnke, Zur Theorie der diophantischen Approximationen, Abh. Math. Sem. Univ. Hamburg 3 (1924), 261-318.
  • 21. E. Bombieri, Algebraic values of meromorphic maps, Invent. Math. 10 (1970), 267-287. MR 306201
  • 22. E. Bombieri and S. Lang, Analytic subgroups of group varieties, Invent. Math. 11 (1970), 1-14. MR 296028
  • 23. A. Brumer, On the units of algebraic number fields, Mathematika 14 (1967), 121-124. MR 36 #3746. MR 220694
  • 24. J. W. S. Cassels, An introduction to Diophantine approximation, Cambridge Tracts in Math. and Math. Phys., no. 45, Cambridge Univ. Press, New York, 1957. MR 19, 396. MR 87708
  • 25. J. Coates, An effective p-adic analogue of a theorem of Thue, Acta Arith. 15 (1968/69), 279-305. MR 39 #4095. MR 242768
  • 26. J. Coates, Construction of rational functions on a curve, Proc. Cambridge Philos. Soc. 68 (1970), 105-123. MR 258831
  • 27. J. Coates, An effective p-adic analogue of a theorem of Thue. II. The greatest prime factor of a binary form, Acta Arith. 17 (1970), 399-412; The Diophantine equation Y2=X3+k, ibid., 425-435. MR 263741
  • 28. J. Coates, The transcendence of linear forms in ω1, ω2, η1, η2, 2πi (to appear).
  • 29. A. Baker and J. Coates, Integer points on curves of genus 1, Proc. Cambridge Philos. Soc. 67 (1970), 595-602. MR 41 #1638. MR 256983
  • 30. R. M. Damerell, L-functions of elliptic curves with multiplication. I, Acta Arith. 17 (1970), 287-301. MR 285540
  • 31. H. Davenport and K. Roth, Rational approximations to algebraic numbers, Mathematika 2 (1955), 160-167. MR 17, 1060. MR 77577
  • 32. H. Davenport and W. Schmidt, Dirichlet's theorem on Diophantine approximations. II, Acta Arith. 17 (1970), 413-424. MR 279040
  • 33. P. Erdös, Some results on Diophantine approximation, Acta Arith. 5 (1959), 359-369. MR 22 #12091. MR 121352
  • 34. N. I. Feldman, Approximation of certain transcendental numbers. I: The approximation of logarithms of algebraic numbers; II: The approximation of certain numbers associated with the Weierstrass function, Izv. Akad. Nauk SSSR Ser. Mat. 15 (1951), 53-74, 153-176; English transl., Amer. Math. Soc. Transl. (2) 59 (1966), 224-270. MR 12, 595; MR 13, 117. MR 39768
  • 35. N. I. Feldman, Simultaneous approximation of the periods of an elliptic function by algebraic numbers, Izv. Akad. Nauk SSSR Ser. Mat. 22 (1958), 563-576; English transl., Amer. Math. Soc. Transl. (2) 59 (1966), 271-284. MR 20 #5895. MR 99456
  • 36. N. I. Feldman, Approximation of the logarithms of algebraic numbers by algebraic numbers, Izv. Akad. Nauk SSSR Ser. Mat. 24 (1960), 475-492; English transl., Amer. Math. Soc. Transl. (2) 58 (1966), 125-142. MR 22 #5623b. MR 114805
  • 37. N. I. Feldman, On the measure of transcendence of π, Izv. Akad. Nauk SSSR Ser. Mat. 24 (1960), 357-368; English transl., Amer. Math. Soc. Transl. (2) 58 (1966), 110-124. MR 22 #5632a.
  • 38. N. I. Feldman, Arithmetic properties of the solutions of a transcendental equation, Vestnik Moskov. Univ. Ser. I Mat. Meh. 1964, no. 1, 13-20; English transl., Amer. Math. Soc. Transl. (2) 66 (1968), 145-153. MR 28 #2091. MR 158869
  • 39. N. I. Feldman, Estimate for a linear form of logarithms of algebraic numbers, Mat. Sb. 76 (118) (1968), 304-319 = Math. USSR Sb. 5 (1968), 291-307. MR 37 #4025. MR 228445
  • 40. N. I. Feldman, Improved estimate for a linear form of the logarithms of algebraic numbers, Mat. Sb. 77 (119) (1968), 423-436 = Math. USSR Sb. 6 (1968), 398-406. MR 38 #1059. MR 232736
  • 41. N. I. Feldman, A certain inequality for a linear form in the logarithms of algebraic numbers, Mat. Zametki 5 (1969), 681-689. (Russian) MR 40 #2610. MR 249365
  • 42. A. O. Gelfond and N. I. Feldman, On the measure of relative transcendentality of certain numbers, Izv. Akad. Nauk SSSR Ser. Mat. 14 (1950), 493-500. (Russian) MR 12, 679. MR 40349
  • 43. A. O. Gelfond, Sur les propriétés arithmétiques des fonctions entières, Tôhoku Math. J. 30 (1929), 280-285.
  • 44. A. O. Gelfond, Transcendental and algebraic numbers, GITTL, Moscow, 1952; English transl., Dover, New York, 1960. MR 15, 292; MR 22 #2598. MR 111736
  • 45. C. Hermite, "Sur la fonction exponentielle," in Oeuvres, Vol. III, pp. 150-181.
  • 46. A. Ya. Khinchin (A. Ja. Hinčin), Continued fractions, Fizmatgiz, Moscow, 1961; English transl., Univ. of Chicago Press, Chicago, Ill., 1964. MR 28 #5037. MR 161833
  • 47. J. F. Koksma, Über die Mahlersche Klasseneinteilung der transzendenten Zahlen und die Approximation komplexer Zahlen durch algebraische Zahlen, Monatsh. Math. Phys. 48 (1939), 176-189. MR 1, 137. MR 845
  • 48. S. Lang, Report on Diophantine approximations, Bull. Soc. Math. France 93 (1965), 177-192. MR 33 #1286. MR 193064
  • 49. S. Lang, Introduction to transcendental numbers, Addison-Wesley, Reading, Mass., 1966. MR 35 #5397. MR 214547
  • 50. S. Lang, Introduction to Diophantine approximations, Addison-Wesley, Reading, Mass., 1966. MR 35 #129. MR 209227
  • 51. S. Lang, Diophantine approximations on toruses, Amer. J. Math. 86 (1964), 521-533. MR 29 #2220. MR 164929
  • 52. S. Lang, Diophantine geometry, Interscience Tracts in Pure and Appl. Math., no. 11, Interscience, New York, 1962. MR 26 #119. MR 142550
  • 53. S. Lang, Algebraic number theory, Addison-Wesley, Reading, Mass., 1970. MR 282947
  • 54. H. W. Leopoldt, Zur Arithmetik in abelschen Zahlkörpern, J. Reine Angew. Math. 209 (1962), 54-71. MR 25 #3034. MR 139602
  • 55. W. J. LeVeque, On the frequency of small fractional parts in certain real sequences. II, Trans. Amer. Math. Soc. 94 (1960), 130-149. MR 22 #12089. MR 121350
  • 56. F. Lindemann, Über die Zahl π, Math. Ann. 20 (1882), 213-225.
  • 57. Kurt Mahler, Über transzendente 𝑃-adische Zahlen, Compositio Math. 2 (1935), 259–275 (German). MR 1556919
  • 58. K. Mahler, Zur Approximation der Exponentialfunktion und des Logarithmus, J. Reine Angew. Math. 66 (1932), 118-150.
  • 59. K. Mahler, On the approximation of logarithms of algebraic numbers, Philos. Trans. Roy. Soc. London Ser. A 245 (1953), 371-398. MR 14, 624. MR 52471
  • 60. K. Mahler, On the approximation of π, Nederl. Akad. Wetensch. Proc. Ser. A. 56 = Indag. Math. 15 (1953), 30-42. MR 14, 957.
  • 61. K. Mahler, Ein Übertragungsprinzip für konvexe Körper, Casopis Pěst. Mat. Fys. 68 (1939), 93-102. MR 1, 202. MR 1242
  • 62. K. Mahler, On compound convex bodies. I, Proc. London Math. Soc. (3) 5 (1955), 358-379. MR 17, 589. MR 74460
  • 63. K. Mahler, Applications of some formulae by Hermite to the approximation of exponentials and logarithms, Math. Ann. 168 (1967), 200-227. MR 34 #5754. MR 205929
  • 64. A. Néron, Quasi-fonctions et hauteurs sur les variétiés abéliennes, Ann. of Math.(2) 82 (1965), 249-331. MR 31 #3424. MR 179173
  • 65. A. Ostrowski, Bemerkungen zur Theorie der Diophantischen Approximationen, Abh. Math. Sem. Univ. Hamburg 1 (1921), 77-98.
  • 66. O. Perron, Die Lehre von den Kettenbrüchen, 2nd ed., Chelsea, New York, 1950. MR 12, 254. MR 37384
  • 67. J. Popken, Sur la nature arithmétique du nombre e, C. R. Acad. Sci. Paris 186 (1928), 1505-1507.
  • 68. J. Popken, Zur Transzendenz von 𝑒, Math. Z. 29 (1929), no. 1, 525–541 (German). MR 1545025, https://doi.org/10.1007/BF01180551
  • 69. J. Popken, Zur Transzendenz von π, Math. Z. 29 (1929), 542-448.
  • 70. K. Ramachandra, Some applications of Kronecker's limit formulas, Ann. of Math. (2) 80 (1964), 104-148. MR 29 #2241. MR 164950
  • 71. D. Ridout, Rational approximations to algebraic numbers, Mathematika 4 (1957), 125-131. MR 20 #32. MR 93508
  • 72. K. Roth, Rational approximations to algebraic numbers, Mathematika 2 (1955), 1-20; corrigendum, 168. MR 17, 242. MR 72182
  • 73. W. Schmidt, A metrical theorem in Diophantine approximation, Canad. J. Math. 12 (1960), 619-631. MR 22 #9482. MR 118711
  • 74. W. Schmidt, Metrical theorems on fractional parts of sequences, Trans. Amer. Math. Soc. 110 (1964), 493-518. MR 28 #3018. MR 159802
  • 75. W. Schmidt, Simultaneous approximation to a basis of a real numberfield, Amer. J. Math. 88 (1966), 517-527. MR 34 #2529. MR 202669
  • 76. W. Schmidt, Simultaneous approximation to algebraic numbers by rationals, Acta Math. 21 (1970), 189-201. MR 268129
  • 77. W. Schmidt, Lectures on Diophantine approximation, University of Colorado, Boulder, Colo., 1970.
  • 78. T. Schneider, Zur Theorie der Abelschen Funktionen und Integrale, J. Reine Angew. Math. 183 (1941), 110-128. MR 3, 266. MR 6170
  • 79. T. Schneider, Ein satz über ganzwertige Funktionen als Prinzip für Transzendenzbeweise, Math. Ann. 121 (1949), 131-140. MR 11, 160. MR 31498
  • 80. T. Schneider, Einführung in die transzendenten Zahlen, Springer-Verlag, Berlin, 1957. MR 19, 252. MR 86842
  • 81. T. Schneider, Über die Approximation algebraischer Zahlen, J. Reine Angew. Math 175 (1936), 182-192.
  • 82. J.-P. Serre, Abelian l-adic representations, Benjamin, New York, 1968. MR 263823
  • 83. J.-P. Serre, "Dependence d'exponentielle p-adiques, " in Séminaire Delange-Pisot-Poitou, 1965/66, Exposé 15, Secrétariat mathématique, Paris, 1967. MR 35 #6507.
  • 84. A. Borel, et al., Seminar on complex multiplication, Lecture Notes in Math., no. 21, Springer-Verlag, Berlin and New York, 1966. MR 34 #1278. MR 201394
  • 85. A. B. Šidlovskĭ, On criteria for algebraic independence of values of a class of integral functions, Izv. Akad. Nauk SSSR Ser. Mat. 23 (1959), 35-66; English transl., Amer. Math. Soc. Transl. (2) 22 (1962), 339-370. MR 21 #1295. MR 102503
  • 86. C. L. Siegel, Über einige Anwendungen diophantischer Approximationen, Abh. Preuss. Akad. Wiss. 1929, 1-41.
  • 87. C. L. Siegel, Transcendental numbers, Ann. of Math. Studies, no. 16, Princeton Univ. Press, Princeton, N. J., 1949. MR 11, 330. MR 32684
  • 88. C. L. Siegel, Bestimmung der elliptischen Modulfunktion durch eine Transformationsgleichung, Abh. Math. Sem. Univ. Hamburg 27 (1964/65), 32-38. MR 29 #2391. MR 165102
  • 89. V. G. Sprindžuk, The irrationality of the values of certain transcendental functions, Izv. Akad. SSSR Ser. Mat. 32 (1968), 93-107 = Math. USSR Izv. 2 (1968), 89-104. MR 36 #5087. MR 222035
  • 90. H, Stark, A historical note on complex quadratic fields with class-number one, Proc. Amer. Math. Soc. 21 (1969), 254-255. MR 38 #5743. MR 237461
  • 91. H, Stark, A transcendence theorem for class number problems, Ann. of Math. (to appear). MR 297715
  • 92. M. Waldschmidt, Independence algebrique des valeurs de la fonction exponentielle, Bull. Soc. Math. France (to appear). MR 399006
  • 93. E. Wirsing, On approximations of algebraic numbers by algebraic numbers of bounded degree, Proc. Sympos. Pure Math., vol. 20, Amer. Math. Soc. Providence, R. I., 1971, pp. 213-247. MR 319929
  • 94. E. Wirsing, Approximation mit algebraischen Zahlen beschränkten Grades, J. Reine Angew. Math. 206 (1960), 67-77, MR 26 #79. MR 142510
  • 95. Churchouse and Muir, Continued fractions, algebraic numbers and modular invariants, J. Inst. Math. Appl. (1969), 318-328. MR 255493
  • 96. S. Lang and H. Trotter, Continued fractions of some algebraic numbers, J. Reine Angew. Math. (to appear). MR 306131
  • 97. H. Stark, An explanation of some exotic continued fractions found by Brillhart (to appear). MR 337801

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DOI: https://doi.org/10.1090/S0002-9904-1971-12761-1

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