Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-05T06:49:22.464Z Has data issue: false hasContentIssue false
  • English
  • Français

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 ([email protected])
Natalia Budarina
Affiliation:
Department of Mathematics, Logic House, National University of Maynooth, Co. Kildare, Republic of Ireland ([email protected])
Detta Dickinson
Affiliation:
Department of Mathematics, Logic House, National University of Maynooth, Co. Kildare, Republic of Ireland ([email protected])
Hugh O'Donnell
Affiliation:
Department of Mathematics, Logic House, National University of Maynooth, Co. Kildare, Republic of Ireland ([email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

1.Beresnevich, V., Dickinson, D. and Velani, S. L., Measure theoretic laws for limsup sets, Memoirs of the American Mathematical Society, Volume 846 (American Mathematical Society, Providence, RI, 2006).Google Scholar
2.Beresnevich, V., Dickinson, D. and Velani, S. L., Diophantine approximation on planar curves and the distribution of rational points, Annals Math. 166 (2007), 367–426.CrossRefGoogle Scholar
3.Bernik, V. I. and Dodson, M. M., Metric Diophantine approximation on manifolds, Cambridge Tracts in Mathematics, Volume 137 (Cambridge University Press, 1999).Google Scholar
4.Budarina, N., Dickinson, D. and Levesley, J., Simultaneous Diophantine approximation on polynomial curves, Mathematika 56 (2010), 77–85.CrossRefGoogle Scholar
5.Bugeaud, Y., Approximation by algebraic numbers, Cambridge Tracts in Mathematics, Volume 160 (Cambridge University Press, 2004).Google Scholar
6.Bugeaud, Y., On simultaneous rational approximation to a real number and its integral powers, Annales Inst. Fourier 60 (2010), 2165–2182.CrossRefGoogle Scholar
7.Bugeaud, Y. and Laurent, M., Exponents of Diophantine approximation and Sturmian continued fractions, Annales Inst. Fourier 55 (2005), 773–804.CrossRefGoogle Scholar
8.Bugeaud, Y. and Laurent, M., Exponents of Diophantine approximation, in Diophantine Geometry, CRM Series, Publications of the Scuola Normale Superiore, Volume 4, pp. 101–121 (Springer, 2007).Google Scholar
9.Güting, R., Zur Berechnung der Mahlerschen Funktionen wn, J. Reine Angew. Math. 232 (1968), 122–135.Google Scholar
10.Mahler, K., Zur Approximation P-adischer Irrationalzahlen, Nieuw Arch. Wisk. 18 (1934), 22–34.Google Scholar
11.Mahler, K., Über eine Klasseneinteilung der P-adischen Zahlen, Mathematica (Zutphen) 3B (1935), 177–185.Google Scholar
12.Mahler, K., Ein Übertragungsprinzip für lineare Ungleichungen, Časopis Pӗst. Mat. Fys. 68 (1939), 85–92.CrossRefGoogle Scholar
13.Melničuk, Ju. V., Hausdorff dimension in Diophantine approximation of p-adic numbers, Ukrain. Mat. Zh. 32 (1980), 118–124, 144 (in Russian).Google Scholar
14.Robert, A. M., A course in p-adic analysis, Graduate Texts in Mathematics, Volume 198 (Springer, 2000).Google Scholar
15.Sprindžuk, V. G., Mahler's problem in metric number theory, Translations of Mathematical Monographs, Volume 25 (American Mathematical Society, Providence, RI, 1969).Google Scholar
16.Vaughan, R. C. and Velani, S., Diophantine approximation on planar curves: the convergence theory, Invent. Math. 166 (2006), 103–124.CrossRefGoogle Scholar