Inhomogeneous Diophantine approximation
with general error functions
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
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
@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 -
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
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