On inhomogeneous Diophantine approximation and the NST-algorithm (Q2707571)

Let \(\theta\) be an irrational number, \(\varphi\) a real number and assume \(q\theta- \varphi\) is not integral for any integer \(q\). Define NEWLINE\[NEWLINE{\mathcal M}_+(\theta,\varphi)= \liminf_{q\to\infty} q\|q\theta- \varphi\|, \qquad {\mathcal M}_-(\theta,\varphi)= \liminf_{q\to-\infty} q\|q\theta- \varphi\|,NEWLINE\]NEWLINE and let \({\mathcal M}(\theta,\varphi)= \min({\mathcal M}_+ (\theta,\varphi), {\mathcal M}_- (\theta,\varphi))\). In this paper the author establishes a relationship between \({\mathcal M}_-(\theta,\varphi)\) and the NST-algorithm of Nishioka, Shiokawa and Tamura, which allows him to determine the exact value of \({\mathcal M}(\theta,\varphi)\) for a certain class of pairs \((\theta,\varphi)\). As a result the author finds that \({\mathcal M}(e,1/2)= 1/8\).NEWLINENEWLINEFor the entire collection see [Zbl 0932.00040].