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

Part of: Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  07 May 2021

Dzmitry Badziahin ,
Yann Bugeaud  and
Johannes Schleischitz
Dzmitry Badziahin
Affiliation:
School of Mathematics and Statistics, The University of Sydney, Camperdown, NSW2006, Australia ([email protected])
Yann Bugeaud
Affiliation:
Université de Strasbourg, Mathématiques, 7, rue René Descartes, Strasbourg67084, France ([email protected])
Johannes Schleischitz
Affiliation:
Middle East Technical University, Northern Cyprus Campus, Kalkanli, Güzelyurt, Turkey ([email protected])
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Abstract

Let $p$ be a prime number. For a positive integer $n$ and a real number $\xi$, let $\lambda _n (\xi )$ denote the supremum of the real numbers $\lambda$ for which there are infinitely many integer tuples $(x_0, x_1, \ldots , x_n)$ such that $| x_0 \xi - x_1|_p, \ldots , | x_0 \xi ^{n} - x_n|_p$ are all less than $X^{-\lambda - 1}$, where $X$ is the maximum of $|x_0|, |x_1|, \ldots , |x_n|$. We establish new results on the Hausdorff dimension of the set of real numbers $\xi$ for which $\lambda _n (\xi )$ is equal to (or greater than or equal to) a given value.

Keywords

simultaneous approximationtransference theorem

MSC classification

Primary: 11J13: Simultaneous homogeneous approximation, linear forms
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 64 , Issue 2 , May 2021 , pp. 317 - 337
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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