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On simultaneous rational approximation to a p-adic number and its integral powers

Part of: Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  17 August 2011

Yann Bugeaud ,
Natalia Budarina ,
Detta Dickinson  and
Hugh O'Donnell
Yann Bugeaud
Affiliation:
UFR de Mathématique et d'Informatique, Université de Strasbourg, 7 rue René Descartes, 67084 Strasbourg, France (bugeaud@math.unistra.fr)
Natalia Budarina
Affiliation:
Department of Mathematics, Logic House, National University of Maynooth, Co. Kildare, Republic of Ireland (ddickinson@maths.nuim.ie)
Detta Dickinson
Affiliation:
Department of Mathematics, Logic House, National University of Maynooth, Co. Kildare, Republic of Ireland (ddickinson@maths.nuim.ie)
Hugh O'Donnell
Affiliation:
Department of Mathematics, Logic House, National University of Maynooth, Co. Kildare, Republic of Ireland (ddickinson@maths.nuim.ie)
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Abstract

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Let p be a prime number. For a positive integer n and a p-adic number ξ, let λn(ξ) denote the supremum of the real numbers λ such that there are arbitrarily large positive integers q such that ‖qξ‖p,‖qξ2‖p,…,‖qξn‖p are all less than q−λ−1. Here, ‖x‖p denotes the infimum of |x−n|p as n runs through the integers. We study the set of values taken by the function λn.

Keywords

Diophantine approximationHausdorff dimensionp-adic number

MSC classification

Secondary: 11J13: Simultaneous homogeneous approximation, linear forms 11J61: Approximation in non-Archimedean valuations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 54 , Issue 3 , October 2011 , pp. 599 - 612
Copyright
Copyright © Edinburgh Mathematical Society 2011

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