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ON SPRINDŽUK’S CLASSIFICATION OF $p$-ADIC NUMBERS

Part of: Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  12 December 2019

YANN BUGEAUD  and
GÜLCAN KEKEÇ
YANN BUGEAUD
Affiliation:
Université de Strasbourg, CNRS, IRMA UMR 7501, 7, rue René Descartes, 67000 Strasbourg, France e-mail: yann.bugeaud@math.unistra.fr
GÜLCAN KEKEÇ
Affiliation:
Istanbul University, Faculty of Science, Department of Mathematics, 34134 Vezneciler, Istanbul, Turkey e-mail: gulkekec@istanbul.edu.tr
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Abstract

We carry Sprindžuk’s classification of the complex numbers to the field $\mathbb{Q}_{p}$ of $p$-adic numbers. We establish several estimates for the $p$-adic distance between $p$-adic roots of integer polynomials, which we apply to show that almost all $p$-adic numbers, with respect to the Haar measure, are $p$-adic $\tilde{S}$-numbers of order 1.

Keywords

Sprindžuk’s classification of the complex numbersp-adic transcendental numberstranscendence measureroots of an integer polynomial in ℂpLiouville’s inequality

MSC classification

Primary: 11J61: Approximation in non-Archimedean valuations
Secondary: 11J82: Measures of irrationality and of transcendence

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 111 , Issue 2 , October 2021 , pp. 221 - 231
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Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by M. Coons

This research was supported by the Scientific Research Projects Coordination Unit of Istanbul University (project number FUA-2018-31152).

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