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Approximations to the logarithms of certain rational numbers

  • A. Baker
Chapter

Abstract

In a recent paper [1] methods were introduced for investigating the accuracy with which certain algebraic numbers may be approximated by rational numbers. It is the main purpose of the present paper to deduce, using similar techniques, results concerning the accuracy with which the natural logarithms of certain rational numbers may be approximated by rational numbers, or, more generally, by algebraic numbers of bounded degree.

Keywords

Positive Integer Linear Form Natural Logarithm Rational Number Fractional Part 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    A. Baker, Rational approximations to certain algebraic numbers, Proc. Lond. Math. Soc. (3)14(1964), pp. 385–398.CrossRefzbMATHGoogle Scholar
  2. [2]
    N. I. Feldman, Approximation by algebraic, numbers to the logarithms of algebraic numbers, Izw. Akad. Nauk. SSSK. ser. mat. 24 (1960), pp. 475–492 (in Russian).Google Scholar
  3. [3]
    K. Mahler, Zur Approximation der Exponentialfunktion und des Logarith mus, Jour, reine, an gew. Maih. 166 (1932), (I) pp. 118–136, (M) pp. 137–150.Google Scholar
  4. [4]
    K. Mahler, On the approximation of logarithms of algebraic numbers, Philos. Trans. Roy. Soe. Loud., ser. A, 245 (1953), pp. 371–398.CrossRefzbMATHMathSciNetGoogle Scholar
  5. [5]
    K. Mahler, On the approximation of n, Proc. Akad. Wetensch. Amst., ser. A, 56 (1953), pp. 30–42.MathSciNetGoogle Scholar
  6. [6]
    D. Morduchai-Boltowskoj, Sur le logarithme d’un nombre algébrique, Comptes liendus Acad. Sei., Paris, 176 (1923), pp. 724–727.Google Scholar
  7. [7]
    Th. Schneider, Einführung in die transzeudoilen Zahlen, Berlin, Göttingeii, Heidelberg, 1957, Kap. 4.CrossRefGoogle Scholar
  8. [8]
    C. L. Siegel, Über einige Anwendungen diop hantise her Approximationen, Abh. Preuss. Akad. Wiss. (1929), No 1.Google Scholar
  9. [9]
    E. Wirsing, Approximation mit algebraischen Zahlen beschrankten Grades. Jour, reine, angew. Math. 206 (1961), pp. 67–77.zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • A. Baker
    • 1
  1. 1.CambridgeUK

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