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

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|...

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Bibliographic Details
Published inarXiv.org
Main Authors Badziahin, Dzmitry, Bugeaud, Yann, Schleischitz, Johannes
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 25.03.2021
Subjects
Integers
Numbers
Prime numbers
Real numbers
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Summary: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.
ISSN:2331-8422