Inhomogeneous Diophantine approximation with general error functions
Volume 160 / 2013
Acta Arithmetica 160 (2013), 25-35
MSC: Primary 28A80; Secondary 37E05, 28A78.
DOI: 10.4064/aa160-1-2
Abstract
Let $\alpha$ be an irrational and $\varphi: \mathbb N \rightarrow \mathbb R^+$ be a function decreasing to zero. Let $\omega(\alpha):= \sup \{\theta \geq 1: \liminf_{n\to \infty}n^{\theta} \|n\alpha\|=0\}$. For any $\alpha$ with a given $\omega(\alpha)$, we give some sharp estimates for the Hausdorff dimension of the set \[ E_{\varphi}(\alpha):=\{y\in \mathbb R: \|n\alpha -y\| < \varphi(n) \text{ for infinitely many } n\}, \] where $\|\cdot\|$ denotes the distance to the nearest integer.
