2024 CLASSICAL AND UNIFORM EXPONENTS OF MULTIPLICATIVE p-ADIC APPROXIMATION
Yann Bugeaud, Johannes Schleischitz
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Publ. Mat. 68(1): 3-26 (2024). DOI: 10.5565/PUBLMAT6812401

Abstract

Let p be a prime number and ξ an irrational p-adic number. Its irrationality exponent μ(ξ) is the supremum of the real numbers μ for which the system of inequalities

0<max{|x|,|y|}X, |yξx|pXμ

has a solution in integers x, y for arbitrarily large real number X. Its multiplicative irrationality exponent μ×(ξ) (resp., uniform multiplicative irrationality exponent μ^×(ξ)) is the supremum of the real numbers μ^ for which the system of inequalities

0<|xy|1/2X, |yξx|pXμ^

has a solution in integers x, y for arbitrarily large (resp., for every sufficiently large) real number X. It is not difficult to show that μ(ξ)μ×(ξ)2μ(ξ) and μ^×(ξ)4. We establish that the ratio between the multiplicative irrationality exponent μ× and the irrationality exponent μ can take any given value in [1, 2]. Furthermore, we prove that μ^×(ξ)(5+5)/2 for every p-adic number ξ.

Acknowledgements

The authors are very grateful to the referees,whose numerous detailed remarks and corrections helped them to considerably improve the presentation of the paper.

Citation

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Yann Bugeaud. Johannes Schleischitz. "CLASSICAL AND UNIFORM EXPONENTS OF MULTIPLICATIVE p-ADIC APPROXIMATION." Publ. Mat. 68 (1) 3 - 26, 2024. https://doi.org/10.5565/PUBLMAT6812401

Information

Received: 26 May 2021; Accepted: 1 October 2021; Published: 2024
First available in Project Euclid: 25 December 2023

MathSciNet: MR4682721
Digital Object Identifier: 10.5565/PUBLMAT6812401

Subjects:
Primary: 11J04 , 11J61

Keywords: exponent of approximation , p-adic number , Rational approximation

Rights: Copyright © 2024 Universitat Autònoma de Barcelona, Departament de Matemàtiques

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