Inhomogeneous Diophantine approximation with general error functions

Lingmin Liao¹ ; Michał Rams²

¹ LAMA UMR 8050, CNRS Université Paris-Est Créteil 61 Avenue du Général de Gaulle 94010 Créteil Cedex, France
² Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-956 Warszawa, Poland

Acta Arithmetica, Tome 160 (2013) no. 1, pp. 25-35.

Résumé

Let $\alpha$ be an irrational and $\phi :\mathbb{N}\to {\mathbb{R}}^{+}$ be a function decreasing to zero. Let $\omega (\alpha ):=sup\{\theta \ge 1:\underset{n\to \mathrm{\infty }}{lim inf}{n}^{\theta }\Vert n\alpha \Vert =0\}$. 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 $\Vert \cdot \Vert$ denotes the distance to the nearest integer.

Détail

DOI : 10.4064/aa160-1-2

Mots-clés : alpha irrational varphi mathbb rightarrow mathbb function decreasing zero omega alpha sup theta geq liminf infty theta alpha alpha given omega alpha sharp estimates hausdorff dimension set varphi alpha mathbb alpha y varphi text infinitely many where cdot denotes distance nearest integer