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A NOTE ON INHOMOGENEOUS DIOPHANTINE APPROXIMATION WITH A GENERAL ERROR FUNCTION

Published online by Cambridge University Press:  23 August 2006

AI-HUA FAN
Affiliation:
Department of Mathematics, Wuhan University, Wuhan, Hubei, 430072, P.R.China and LAMFA, CNRS UMR 6140, Université de Picardie, 80039 Amiens, France e-mail: aihua.fan@u-picardie.fr
JUN WU
Affiliation:
Department of Mathematics, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, P.R.China and LAMFA, CNRS UMR 6140, Université de Picardie, 80039 Amiens, France e-mail: wujunyu@public.wh.hb.cn
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Abstract

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Let $\alpha$ be an irrational number and $\varphi$: $\mathbb{N} \to \mathbb{R^+}$ be a decreasing sequence tending to zero. Consider the set \[E_{\varphi}(\alpha)=\{\beta \in \mathbb{R}: \ \|n \alpha- \beta\|<\varphi(n)\ {\rm {holds\ for\ infinitely\ many}} \ n \in \mathbb{N}\}\], where $\|{\cdot}\|$ denotes the distance to the nearest integer. We show that for general error function $\varphi$, the Hausdorff dimension of $E_{\varphi}(\alpha)$ depends not only on $\varphi$, but also heavily on $\alpha$. However, recall that the Hausdorff dimension of $E_{\varphi}(\alpha)$ is independent of $\alpha$ when $\varphi(n) = n^{-\gamma}$ with $\gamma >1$.

Keywords

Type
Research Article
Copyright
2006 Glasgow Mathematical Journal Trust