More on inhomogeneous Diophantine approximation
Given irrational real numbers \(\alpha\) and a real number \(\gamma\) in the unit interval, define the one-sided approximation constants \[ M_+ (\alpha,\gamma)= \liminf_{n\to-\infty} n\|n\alpha- \gamma\|, \qquad M_-(\alpha,\gamma)= \liminf_{n\to_\infty} n\|n\alpha-\gamma\|, \] and the inhomogeneous function \(M(\alpha,\gamma)= \min\{M_+ (\alpha,\gamma), M_- (\alpha,\gamma)\}\), where \(\|x\|\) denotes the distance from \(x\) to the nearest integer. Given any (fixed) irrational \(\alpha\), one is interested in the spectrum \({\mathcal S}(\alpha)\) of values \(M(\alpha,\gamma)\) where \(\gamma\) runs through the real numbers not of the form \(u+\alpha v\) \((u,v\in \mathbb{Z})\). Similarly one defines spectra \({\mathcal S}_+ (\alpha)\), \({\mathcal S}_-(\alpha)\) involving the functions \(M_+\) and \(M_-\), respectively. The author studies these spectra for various classes of numbers \(\alpha\). He works with the backward continued fraction expansion obtained by iterating the singular shift \(T:x\mapsto \lceil 1/x\rceil- 1/x\), and with a kind of \(\alpha\)-expansion of \(\gamma\) [for related work, see \textit{T. Komatsu}, Acta Arith. 86, 305-324 (1998; Zbl 0930.11049) and \textit{T. W. Cusick, A. M. Rockett} and \textit{P. Szüsz}, J. Number Theory 48, 259-283 (1994; Zbl 0820.11042)].
scientific article; zbMATH DE number 1770785