On simultaneous rational approximation to a $p$-adic number and its integral powers, II

Abstract

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 $(x_0,x_1,\dots ,x_n)$ such that $|x_0\xi -x_1{|}_p,\dots ,|x_0{\xi }^n-x_n{|}_p$ are all less than $X^{-\lambda -1}$, where $X$ is the maximum of $|x_0|,|x_1|,\dots ,|x_n|$. 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.

19 pages. arXiv admin note: text overlap with arXiv:1906.05508