  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
arbitrarily large positive integers $q$ such that $|| q \xi ||, || q \xi^2 ||,
\ldots , ||q \xi^n||$ are all less than $q^{-\lambda}$. Here, $|| \cdot ||$
denotes the distance to the nearest integer. We establish new results on the
Hausdorff dimension of the set of real numbers $\xi$ such that $\lambda_n
(\xi)$ is equal (or greater than or equal) to a given value.
