On simultaneous rational approximation to a $p$-adic number and its integral powers, II
| Autor: | Badziahin, Dzmitry, Bugeaud, Yann, Schleischitz, Johannes |
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| Rok vydání: | 2020 |
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| Zdroj: | Proc. Edinb. Math. Soc. 64, no. 2 (2021), 317-337 |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 19 pages. arXiv admin note: text overlap with arXiv:1906.05508 |
| Databáze: | arXiv |
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