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

TL;DR

This paper investigates the Hausdorff dimension of real numbers with specific approximation properties related to p-adic rational approximations of a number and its powers, providing new bounds and results.

Contribution

It establishes new results on the Hausdorff dimension of sets of real numbers with prescribed p-adic approximation exponents.

Findings

New bounds on Hausdorff dimension for approximation sets

Characterization of numbers with large p-adic approximation exponents

Extension of previous approximation theory results

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 $(x_0, x_1, \ldots , x_n)$ 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.