  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.
