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Simultaneous p-adic Diophantine approximation

Part of: Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  31 January 2023

V. BERESNEVICH ,
J. LEVESLEY  and
B. WARD
V. BERESNEVICH
Affiliation:
Department of Mathematics, University of York, Heslington, York, YO10 5DD. e-mails: victor.beresnevich@york.ac.uk, jason.levesley@york.ac.uk, ward.ben1994@gmail.com
J. LEVESLEY
Affiliation:
Department of Mathematics, University of York, Heslington, York, YO10 5DD. e-mails: victor.beresnevich@york.ac.uk, jason.levesley@york.ac.uk, ward.ben1994@gmail.com
B. WARD
Affiliation:
Department of Mathematics, University of York, Heslington, York, YO10 5DD. e-mails: victor.beresnevich@york.ac.uk, jason.levesley@york.ac.uk, ward.ben1994@gmail.com
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Abstract

The aim of this paper is to develop the theory of weighted Diophantine approximation of rational numbers to p-adic numbers. Firstly, we establish complete analogues of Khintchine’s theorem, the Duffin–Schaeffer theorem and the Jarník–Besicovitch theorem for ‘weighted’ simultaneous Diophantine approximation in the p-adic case. Secondly, we obtain a lower bound for the Hausdorff dimension of weighted simultaneously approximable points lying on p-adic manifolds. This is valid for very general classes of curves and manifolds and have natural constraints on the exponents of approximation. The key tools we use in our proofs are the Mass Transference Principle, including its recent extension due to Wang and Wu in 2019, and a Zero-One law for weighted p-adic approximations established in this paper.

MSC classification

Primary: 11J83: Metric theory
Secondary: 11J61: Approximation in non-Archimedean valuations
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 175 , Issue 1 , July 2023 , pp. 13 - 50
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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