More on inhomogeneous Diophantine approximation. (English) Zbl 1014.11043

Pinner, Christopher G.

J. Théor. Nombres Bordx. 13, No. 2, 539-557 (2001).

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 T. Komatsu, Acta Arith. 86, 305-324 (1998; Zbl 0930.11049) and T. W. Cusick, A. M. Rockett and P. Szüsz, J. Number Theory 48, 259-283 (1994; Zbl 0820.11042)].

Reviewer: Gerhard Ramharter (Wien)

MSC:

11J20	Inhomogeneous linear forms
11J70	Continued fractions and generalizations

Keywords:

inhomogeneous approximation constants; semi-regular continued fraction; one-sided approximation spectrum; backward continued fraction expansion; \(\alpha\)-expansion

References:

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