1LAMA UMR 8050, CNRS Université Paris-Est Créteil 61 Avenue du Général de Gaulle 94010 Créteil Cedex, France 2Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-956 Warszawa, Poland
Acta Arithmetica, Tome 160 (2013) no. 1, pp. 25-35
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.
Keywords:
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
Affiliations des auteurs :
Lingmin Liao
1
;
Michał Rams
2
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
@article{10_4064_aa160_1_2,
author = {Lingmin Liao and Micha{\l} Rams},
title = {Inhomogeneous {Diophantine} approximation
with general error functions},
journal = {Acta Arithmetica},
pages = {25--35},
year = {2013},
volume = {160},
number = {1},
doi = {10.4064/aa160-1-2},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/aa160-1-2/}
}
TY - JOUR
AU - Lingmin Liao
AU - Michał Rams
TI - Inhomogeneous Diophantine approximation
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JO - Acta Arithmetica
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with general error functions
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Lingmin Liao; Michał Rams. Inhomogeneous Diophantine approximation
with general error functions. Acta Arithmetica, Tome 160 (2013) no. 1, pp. 25-35. doi: 10.4064/aa160-1-2
