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.

中文翻译：

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关于 p 进数及其积分幂的同时有理逼近，II

让$p$是一个素数。对于一个正整数$n$和一个实数$\xi$， 让$\lambda_n (\xi)$表示实数的上限值$\lambda$有无穷多个整数元组$(x_0, x_1, \ldots , x_n)$这样$| x_0 \xi - x_1|_p, \ldots , | x_0 \xi ^{n} - x_n|_p$都小于$X^{-\lambda - 1}$， 在哪里$X$是最大值$|x_0|, |x_1|, \ldots , |x_n|$. 我们在实数集的 Hausdorff 维数上建立了新的结果$\xi$为此$\lambda_n (\xi)$等于（或大于或等于）给定值。