On inhomogeneous diophantine approximation with some quasi-periodic expressions (Q1586360)

For a real irrational number \(\theta\) and a real \(\phi\) such that \(q\theta -\phi\notin{\mathbb Z}\) for \(q\in{\mathbb Z}\) one defines the inhomogeneous approximation constant \(M(\theta ,\phi):=\liminf_{|q|\to\infty} |q|\cdot \|q\theta -\phi \|\), where \(\|\cdot \|\) denotes the distance to the nearest integer. The author shows that for positive integers \(s\), \(M(e^{1/s} ,{1\over 2}(e^{1/s}-1))=0\), \(M(e^{1/s},{1\over 2})={1\over 18}\), \(M(e^{1/s},{1\over 3})=0\) if \(s\equiv 2\pmod 3\) and \(M(e^{1/s},{1\over 3})={1\over 18}\) if \(s\not\equiv 2\) (mod \(3\)). The ingredients going into the proof are first the continued fraction expansion \(e^{1/s}=\break[1;s-1,1,1,3s-1,1,1,5s-1,1,\ldots ]\) and second a result by the author from a previous paper [Acta Arith. 86, 305-324 (1998; Zbl 0930.11049)], based on ideas of \textit{K. Nishioka, I. Shiokawa}, and \textit{J. Tamura} [J. Number Theory 42, 61-87 (1992; Zbl 0770.11039)], which enables one to compute \(M(\theta ,\phi)\) once the continued fraction expansion of \(\theta\) is known.