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

TL;DR

This paper investigates the Hausdorff dimension of real numbers with specific simultaneous approximation properties to a number and its powers, advancing understanding of their approximation exponents.

Contribution

It provides new results on the Hausdorff dimension of sets of real numbers with prescribed simultaneous approximation exponents to a number and its powers.

Findings

Determines the Hausdorff dimension for sets where the approximation exponent exceeds a threshold.

Establishes bounds and exact values for the dimension related to $eta$-approximable numbers.

Extends previous work on rational approximation to include higher powers and Hausdorff dimension analysis.

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 $|| q \xi ||, || q \xi^2 ||, \ldots , ||q \xi^n||$ 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.