Uniform Diophantine approximation related to beta-transformations

TL;DR

This paper determines the Hausdorff dimension of points in [0,1] that approximate a fixed point under beta-transformations at a specified rate, extending previous results to all points in the interval.

Contribution

It generalizes the Hausdorff dimension calculation for uniform Diophantine approximation in beta-transformations to any point in [0,1], broadening prior work.

Findings

Calculated Hausdorff dimension for approximation sets

Extended previous results to all points in [0,1]

Provided a comprehensive framework for beta-transformation approximation

Abstract

For any $\beta>1$, let $T_\beta$ be the classical $\beta$-transformations. Fix $x_0\in[0,1]$ and a nonnegative real number $\hat{v}$, we compute the Hausdorff dimension of the set of real numbers $x\in[0,1]$ with the property that, for every sufficiently large integer $N$, there is an integer $n$ with $1\leq n\leq N$ such that the distance between $T_\beta^nx$ and $x_0$ is at most equal to $\beta^{-N\hat{v}}$. This work extends the result of Bugeaud and Liao \cite{YLiao2016} to every point $x_0$ in unit interval.