Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-05T19:53:44.119Z Has data issue: false hasContentIssue false
  • English
  • Français

Simultaneous diophantine approximations and Hermite's method

Published online by Cambridge University Press:  17 April 2009

Alain Durand
Alain Durand
Affiliation:
UER des Sciences de Limoges, Départment de Mathématiques, 123 rue Albert Thomas, 87100 Limoges, France.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we generalize a result of Mahler on rational approximations of the exponential function at rational points by proving the following theorem: let n ε N* and αl, …, αn be distinct non-zero rational numbers; there exists a constant c = c(n, αl, …, αn) ≥ 0 such that

for every non-zero integer point (qo, ql, …, qn)and q = max {|ql|, … |qn|, 3}.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 21 , Issue 3 , June 1980 , pp. 463 - 470
Copyright
Copyright © Australian Mathematical Society 1980

References

[1]Baker, A., “On some Diophantine inequalities involving the exponential function”, Canad. J. Math. 17 (1965), 616–626.CrossRefGoogle Scholar
[2]Baker, Alan, Transcendental number theory (Cambridge University Press, Cambridge, 1975).CrossRefGoogle Scholar
[3]Cassels, J.W.S., An introduction to Diophantine approximation (Cambridge Tracts in Mathematics and Mathematical Physics, 45. Cambridge University Press, Cambridge, 1957).Google Scholar
[4]Durand, Alain, “Note on rational approximations of the exponential function at rational points”, Bull. Austral. Math. Soc. 14 (1976), 449–455.CrossRefGoogle Scholar
[5] Н.И. Фельдман, А.Б. Шндловский [Fel'dman, N.I. and Shidlovskii, A.B.], “Развитие и современное состояние теории трансцендентных чисел” [The development and present state of the theory of transcendental numbers], Uspehi Mat. Nauk 22 (1967), No. 3, 3–81; Russian Math. Surveys 22 (1967), No. 3, 1–79.Google Scholar
[6]Mahler, Kurt, “Zur Approximation der Exponentialfunktion und des Logarithmus. I”, J. reine angew. Math. 166 (1932), 118–136.CrossRefGoogle Scholar
[7]Mahler, Kurt, “Zur Approximation der Exponentialfunktion und des Logarithmus. II”, J. reine angew. Math. 166 (1932), 137–150.CrossRefGoogle Scholar
[8]Mahler, Kurt, “On rational approximations of the exponential function at rational points”, Bull. Austral. Math. Soc. 10 (1974), 325–335.CrossRefGoogle Scholar
[9]Mahler, Kurt, “On a paper by A. Baker on the approximation of rational powers of e”, Acta Arith. 27 (1975), 61–87.CrossRefGoogle Scholar
[10]Mahler, Kurt, Lectures on transcendental numbers (Lecture Notes in Mathematics, 546. Springer-Verlag, Berlin, Heidelberg, New York, 1976).CrossRefGoogle Scholar
[11]Popken, J., “Zur Transzendenz von e”, Math. Z. 29 (1929), 525–541.CrossRefGoogle Scholar