Dirichlet uniformly well-approximated numbers

TL;DR

This paper investigates the Hausdorff dimension of Dirichlet uniformly well-approximated numbers, revealing its dependence on the Diophantine properties of a fixed irrational number and identifying a unique discontinuity at =1.

Contribution

It provides the Hausdorff dimension for these sets for any >0 and shows its dependence on the Diophantine nature of , with a notable discontinuity at =1.

Findings

Hausdorff dimension depends on Diophantine properties of 

Dimension is discontinuous at =1

Explicit dimension formulas for all >0

Abstract

Fix an irrational number $\theta$. For a real number $\tau >0$, consider the numbers $y$ satisfying that for all large number $Q$, there exists an integer $1\leq n\leq Q$, such that $\|n\theta-y\|<Q^{-\tau}$, where $\|\cdot\|$ is the distance of a real number to its nearest integer. These numbers are called Dirichlet uniformly well-approximated numbers. For any $\tau>0$, the Haussdorff dimension of the set of these numbers is obtained and is shown to depend on the Diophantine property of $\theta$. It is also proved that with respect to $\tau$, the only possible discontinuous point of the Hausdorff dimension is $\tau=1$.