Classical and uniform exponents of multiplicative p -adic approximation
Bugeaud, Yann (Université de Strasbourg)
Schleischitz, Johannes (Middle East Technical University (Güzelyurt, Turquia))

Date:	2024
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, |yξ − x|p ≤ X−µ has a solution in integers x, y for arbitrarily large real number X. Its multiplicative irrationality exponent µ×(ξ) (resp. , uniform multiplicative irrationality exponent µb×(ξ)) is the supremum of the real numbers µb for which the system of inequalities 0 < |xy| 1/2 ≤ X, |yξ − x|p ≤ X−µb 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 µb×(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 µb×(ξ) ≤ (5 + √ 5)/2 for every p-adic number ξ.
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió publicada
Subject:	Rational approximation ; P-adic number ; Exponent of approximation
Published in:	Publicacions matemàtiques, Vol. 68 Núm. 1 (2024) , p. 3-26 (Articles) , ISSN 2014-4350

Adreça original: https://raco.cat/index.php/PublicacionsMatematiques/article/view/422904
DOI: 10.5565/PUBLMAT6812401

24 p, 423.5 KB

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