On inhomogeneous diophantine approximation and the Nishioka-Shiokawa-Tamura algorithm. (English) Zbl 0930.11049

Komatsu, Takao

Acta Arith. 86, No. 4, 305-324 (1998).

Given a real \(\varphi\) and an irrational \(\vartheta\) such that \(q\vartheta-\varphi\) is not integral for any integer \(q\), let \({\mathcal M}(\vartheta,\varphi)=\lim\inf_{|q|\to\infty}(q\|q\vartheta-\varphi\|)\) be the inhomogeneous approximation constant for the pair \(\vartheta,\varphi\). The author uses an algorithm by K. Nishioka, I. Shiokawa and J. Tamura [J. Number Theory 42, 61-87 (1992; Zbl 0770.11039)] which represents \(\varphi\) in terms of the continued fraction expansion of \(\vartheta\) to evaluate \({\mathcal M}(\vartheta,\varphi)\) for certain classes of pairs such as \(\vartheta=(\sqrt{ab(ab+4)}-ab)/(2a)\) and \(\varphi=1/a\) or \(\varphi=1/\sqrt{ab(ab+4)}\). (For earlier work by the author on the same subject see [J. Number Theory 62, 192-212 (1997; Zbl 0878.11029)]).

Reviewer: Gerhard Ramharter (Wien)

MSC:

11J20	Inhomogeneous linear forms
11J70	Continued fractions and generalizations

Keywords:

Nishioka-Shiokawa-Tamura algorithm; inhomogeneous approximation constant; continued fraction expansion