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IRRATIONALITY AND NONQUADRATICITY MEASURES FOR LOGARITHMS OF ALGEBRAIC NUMBERS

Part of: Diophantine approximation, transcendental number theory Miscellaneous topics of analysis in the complex domain

Published online by Cambridge University Press:  22 November 2012

RAFFAELE MARCOVECCHIO  and
CARLO VIOLA
RAFFAELE MARCOVECCHIO
Affiliation:
Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, 1090 Wien, Austria (email: marcovr8@univie.ac.at)
CARLO VIOLA*
Affiliation:
Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy (email: viola@dm.unipi.it)
*
✉For correspondence; e-mail: viola@dm.unipi.it
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Abstract

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Let 𝕂⊂ℂ be a number field. We show how to compute 𝕂-irrationality measures of a number ξ∉𝕂, and 𝕂-nonquadraticity measures of ξ if [𝕂(ξ):𝕂]>2. By applying the saddle point method to a family of double complex integrals, we prove ℚ(α)-irrationality measures and ℚ(α)-nonquadraticity measures of log α for several algebraic numbers α∈ℂ, improving earlier results due to Amoroso and the second-named author.

Keywords

irrationality measurenonquadraticity measureWeil heightsaddle point method

MSC classification

Secondary: 11J82: Measures of irrationality and of transcendence 11J72: Irrationality; linear independence over a field 30E20: Integration, integrals of Cauchy type, integral representations of analytic functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 92 , Issue 2 , April 2012 , pp. 237 - 267
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

References

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