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Quantitative transcendence results for numbers associated with Liouville numbers. (English) Zbl 0802.11028

This paper provides qualitative results of diophantine approximation related to transcendental complex numbers \(\alpha\) which are well approximated by algebraic numbers. For such numbers \(\alpha\), it is known that \(\alpha^ \beta\) (for \(\beta\) any irrational algebraic number) as well as \(e^ \alpha\) are transcendental numbers; also (under a stronger assumption of good approximation for \(\alpha\)), each of the pairs \((\alpha, \alpha^ \beta)\) and \((\alpha, e^ \alpha)\) consists of two algebraically independent numbers [see for instance, N. I. Fel’dman, Vestn. Mosk. Univ. Ser. I 19, 13-20 (1964; Zbl 0124.281)].
Here the author provides quantitative versions of these results, namely transcendence measures for \(\alpha^ \beta\) (for \(\beta\) any irrational algebraic number) and \(e^ \alpha\), as well as measures of algebraic independence for \((\alpha, \alpha^ \beta)\) and \((\alpha, e^ \alpha)\). The main tool is a lower bound for linear forms in the logarithms of algebraic numbers.

MSC:

11J85 Algebraic independence; Gel’fond’s method
11J82 Measures of irrationality and of transcendence
11J86 Linear forms in logarithms; Baker’s method

Citations:

Zbl 0124.281
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Full Text: DOI

References:

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