On simultaneous rational approximation to a real number and its integral powers, II

Abstract

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 $||q\xi ||,||q{\xi }^2||,\dots ,||q{\xi }^n||$ 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.

17 pages