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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Published in | arXiv.org |
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Main Authors | , , |
Format | Paper |
Language | English |
Published |
Ithaca
Cornell University Library, arXiv.org
25.03.2021
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Subjects | |
Online Access | Get full text |
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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. |
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ISSN: | 2331-8422 |