×

zbMATH — the first resource for mathematics

On inhomogeneous continued fraction expansions and inhomogeneous diophantine approximation. (English) Zbl 0878.11029
Given any pair of real numbers \(\vartheta,\varphi\), define the inhomogeneous approximation constants as \[ {\mathcal M}(\vartheta,\varphi)= \min\{{\mathcal M}_+(\vartheta,\varphi), {\mathcal M}_-(\vartheta,\varphi)\}, \] where \({\mathcal M}_+(\vartheta,\varphi)= \liminf_{q\to\infty} (q|q\vartheta- \varphi|)\) and \({\mathcal M}_-(\vartheta,\varphi)= {\mathcal M}_+(\vartheta,-\varphi)\). There are several known algorithms for inhomogeneous diophantine approximation which have been or can be used to deal with these approximation constants [see J. W. S. Cassels, Math. Ann. 127, 288-304 (1954; Zbl 0055.04401); R. Descombes, Ann. Sci. Éc. Norm. Supér., III. Sér. 73, 283-355 (1956; Zbl 0072.03802); V. Turán-Sos, Acta Math. Acad. Sci. Hung. 9, 229-241 (1958; Zbl 0086.03902); K. Nishioka, I. Shiokawa and J.-I. Tamura, J. Number Theory 42, 61-87 (1992; Zbl 0770.11039); S. Itô and H. Tachii, Tokyo J. Math. 16, 261-289 (1993; Zbl 0795.11027); J. M. Borwein and P. B. Borwein, J. Number Theory 43, 293-318 (1993; Zbl 0778.11039); T. W. Cusick, A. Rockett and P. Szüsz, J. Number Theory 48, 259-283 (1994; Zbl 0820.11042)]. Each of these algorithms expresses \(\varphi\), in some way, in terms of the approximation denominators \(q_n\) provided by the ordinary continued fraction expansion of \(\vartheta\).
The author exhibits a new algorithm which allows in principle for the computation of the above constants. He proves the formula \[ {\mathcal M}_+(\vartheta,\varphi)= \liminf_{n\to\infty} (k_n|k_n\vartheta-\varphi|) \] where \(k_n\) is an integer which realises the minimal value of \(|k\vartheta-\varphi|\), \(0\leq k<q_n\). He uses this for the effective computation of \({\mathcal M}(\vartheta^*,1/a)\), \({\mathcal M}(\vartheta^*,1/(2a))\), and \({\mathcal M}(\vartheta^*,1/(a^2+4)^{1/2})\) when \(\vartheta^*= ((a^2+4)^{1/2}-a)/2\).
Reviewer: G.Ramharter (Wien)

MSC:
11J70 Continued fractions and generalizations
PDF BibTeX XML Cite
Full Text: DOI