Inhomogeneous Diophantine approximation with general error functions

Volume 160 / 2013

Lingmin Liao, Michał Rams Acta Arithmetica 160 (2013), 25-35 MSC: Primary 28A80; Secondary 37E05, 28A78. DOI: 10.4064/aa160-1-2


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.


  • Lingmin LiaoLAMA UMR 8050, CNRS
    Université Paris-Est Créteil
    61 Avenue du Général de Gaulle
    94010 Créteil Cedex, France
  • Michał RamsInstitute of Mathematics
    Polish Academy of Sciences
    Śniadeckich 8
    00-956 Warszawa, Poland

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