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

Part of: Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  30 July 2018

YANN BUGEAUD Open the ORCID record for YANN BUGEAUD [Opens in a new window]  and
GÜLCAN KEKEÇ Open the ORCID record for GÜLCAN KEKEÇ [Opens in a new window]
YANN BUGEAUD*
Affiliation:
Université de Strasbourg, CNRS, IRMA UMR 7501, 7, rue René Descartes, 67000 Strasbourg, France email [email protected]
GÜLCAN KEKEÇ
Affiliation:
Department of Mathematics, Faculty of Science, Istanbul University, 34134, Vezneciler-Istanbul, Turkey email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

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We give transcendence measures for $p$-adic numbers $\unicode[STIX]{x1D709}$, having good rational (respectively, integer) approximations, that force them to be either $p$-adic $S$-numbers or $p$-adic $T$-numbers.

Keywords

Mahler’s classification of p-adic numbersp-adic S-numberp-adic T-numbertranscendence measure

MSC classification

Primary: 11J61: Approximation in non-Archimedean valuations
Secondary: 11J82: Measures of irrationality and of transcendence
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 98 , Issue 2 , October 2018 , pp. 203 - 211
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

This work is supported by the Scientific Research Projects Coordination Unit of Istanbul University, project number IRP-52249.

References

Adamczewski, B. and Bugeaud, Y., ‘Mesures de transcendance et aspects quantitatifs de la méthode de Thue–Siegel–Roth–Schmidt’, Proc. Lond. Math. Soc. 101 (2010), 1–26.Google Scholar
Baker, A., ‘On Mahler’s classification of transcendental numbers’, Acta Math. 111 (1964), 97–120.Google Scholar
Bugeaud, Y., Approximation by Algebraic Numbers, Cambridge Tracts in Mathematics, 160 (Cambridge University Press, Cambridge, 2004).Google Scholar
Bugeaud, Y., ‘On the b-ary expansion of an algebraic number’, Rend. Semin. Mat. Univ. Padova 118 (2007), 217–233.Google Scholar
Bugeaud, Y. and Evertse, J.-H., ‘On two notions of complexity of algebraic numbers’, Acta Arith. 133 (2008), 221–250.Google Scholar
Evertse, J.-H., ‘The number of algebraic numbers of given degree approximating a given algebraic number’, in: Analytic Number Theory (Kyoto, 1996), London Mathematical Society Lecture Note Series, 247 (Cambridge University Press, Cambridge, 1997), 53–83.Google Scholar
Koksma, J. F., ‘Über die Mahlersche Klasseneinteilung der transzendenten Zahlen und die Approximation komplexer Zahlen durch algebraische Zahlen’, Monatsh. Math. Phys. 48 (1939), 176–189.Google Scholar
Locher, H., ‘On the number of good approximations of algebraic numbers by algebraic numbers of bounded degree’, Acta Arith. 89 (1999), 97–122.Google Scholar
Mahler, K., ‘Zur Approximation der Exponentialfunktionen und des Logarithmus. I, II’, J. reine angew. Math. 166 (1932), 118–150.Google Scholar
Mahler, K., ‘Über eine Klasseneinteilung der p-adischen Zahlen’, Mathematica (Leiden) 3 (1935), 177–185.Google Scholar
Ridout, D., ‘The p-adic generalization of the Thue–Siegel–Roth theorem’, Mathematika 5 (1958), 40–48.Google Scholar
Roth, K. F., ‘Rational approximations to algebraic numbers’, Mathematika 2 (1955), 1–20; corrigendum, 168.Google Scholar
Schlickewei, H. P., ‘ p-adic T-numbers do exist’, Acta Arith. 39 (1981), 181–191.Google Scholar