Geodesic
Inhomogeneous Diophantine approximation with general error functions
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
Acta Arithmetica, Tome 160 (2013) no. 1, pp. 25-35 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

Voir la notice de l'article

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.
DOI : 10.4064/aa160-1-2
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

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
 with general error functions
JO  - Acta Arithmetica
PY  - 2013
SP  - 25
EP  - 35
VL  - 160
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4064/aa160-1-2/
DO  - 10.4064/aa160-1-2
LA  - en
ID  - 10_4064_aa160_1_2
ER  - 
%0 Journal Article
%A Lingmin Liao
%A Michał Rams
%T Inhomogeneous Diophantine approximation
 with general error functions
%J Acta Arithmetica
%D 2013
%P 25-35
%V 160
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4064/aa160-1-2/
%R 10.4064/aa160-1-2
%G en
%F 10_4064_aa160_1_2
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

Cité par Sources :