A NOTE ON INHOMOGENEOUS DIOPHANTINE APPROXIMATION WITH A GENERAL ERROR FUNCTION

Abstract

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 $\gamma >1$.