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Note on rational approximations of the exponential function at rational points

Published online by Cambridge University Press:  17 April 2009

Alain Durand
Alain Durand
Affiliation:
Département de Mathématiques, U.E.R. des Sciences de Limoges, Limoges, France.
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Abstract

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Let p, q, u, and v be any four positive integers, and let δ be a number in the interval 0 < δ ≤ 2. In one of his papers, Kurt Mahler, Bull. Austral. Math. Soc. 10 (1974), 325–335, proved that if q satisfies the inequalities

then

In this note, by a slightly different treatment of some inequalities in Mahler's paper, we easily obtain the same result with q only restricted by the first condition.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 14 , Issue 3 , June 1976 , pp. 449 - 455
Copyright
Copyright © Australian Mathematical Society 1976

References

[1]Bundschuh, Peter, “Irrationalitätsmaße für e α, α # 0 rational oder Liouville-Zahl”, Math. Ann. 192 (1971), 229–242.CrossRefGoogle Scholar
[2]Mahler, Kurt, “On rational approximations of the exponential function at rational points”, Bull. Austral. Math. Soc. 10 (1974), 325–335CrossRefGoogle Scholar