Inhomogeneous Diophantine approximation with general error functions (Q2840494)

For irrational numbers \(\alpha\), the authors consider the Hausdorff dimension \(\dim_{H}\) of the set NEWLINE\[NEWLINE E_{\phi}(\alpha):= \{ y: \|n\alpha - y\| < \phi(n) \text{ for infinitely many } n\},NEWLINE\]NEWLINE where \(\phi: {\mathbb N} \to {\mathbb R}^{+}\) decreases to zero and \(\| \, \cdot \, \|\) denotes the distance to the nearest integer. The main results of the paper are stated in the following two theorems, where NEWLINE\[NEWLINE\omega(\alpha):= \sup \{\theta \geq 1: \liminf n^{\theta}\|n\alpha\| =0\},NEWLINE\]NEWLINE NEWLINE\[NEWLINE\lambda(\phi):= \liminf \log \phi(n)/\log n, \kappa(\phi):=\limsup \log \phi(n)/\log n,NEWLINE\]NEWLINE and \(u_{\phi} :=1/\lambda(1/\phi)\) as well as \( l_{\phi}:= 1/\kappa(1/\phi)\).NEWLINENEWLINETheorem 1. For each irrational \(\alpha\) and \( 0 \leq l < u \leq 1\) with \(u > 1/\omega(\alpha)\), there exists a decreasing function \(\phi: {\mathbb N} \to {\mathbb R}^{+}\) with \(l_{\phi} =l \) and \(u_{\phi} = u\) such that NEWLINE\[NEWLINE \dim_{H} E_{\phi}(\alpha) = \max \left\{l, \frac{1+u}{1+\omega(\alpha)}\right\} < u.NEWLINE\]NEWLINE { Theorem 2.} For \( 0 \leq l < u \leq 1\), there exists a decreasing function \(\phi: {\mathbb N} \to {\mathbb R}^{+}\) with \(l_{\phi} =l \) and \(u_{\phi} = u\) such that for each non-Liouville number \(\alpha\), we have NEWLINE\[NEWLINE \dim_{H} E_{\phi}(\alpha) =u.NEWLINE\]