On inhomogeneous Diophantine approximation and the Nishioka-Shiokawa-Tamura algorithm
Takao Komatsu
Acta Arithmetica, Tome 84 (1998), p. 305-324 / Harvested from The Polish Digital Mathematics Library

We obtain the values concerning (θ,ϕ)=liminf|q||q|qθ-ϕ using the algorithm by Nishioka, Shiokawa and Tamura. In application, we give the values (θ,1/2), (θ,1/a), (θ,1/√(ab(ab+4))) and so on when θ = (√(ab(ab+4)) - ab)/(2a) = [0;a,b,a,b,...].

Publié le : 1998-01-01
EUDML-ID : urn:eudml:doc:207199
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     author = {Takao Komatsu},
     title = {On inhomogeneous Diophantine approximation and the Nishioka-Shiokawa-Tamura algorithm},
     journal = {Acta Arithmetica},
     volume = {84},
     year = {1998},
     pages = {305-324},
     zbl = {0930.11049},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-aav86i4p305bwm}
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Takao Komatsu. On inhomogeneous Diophantine approximation and the Nishioka-Shiokawa-Tamura algorithm. Acta Arithmetica, Tome 84 (1998) pp. 305-324. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-aav86i4p305bwm/

[000] [1] J. W. S. Cassels, Über l̲imx+x|ϑx+α-y|, Math. Ann. 127 (1954), 288-304.

[001] [2] T. W. Cusick, A. M. Rockett and P. Szüsz, On inhomogeneous Diophantine approximation, J. Number Theory 48 (1994), 259-283. | Zbl 0820.11042

[002] [3] R. Descombes, Sur la répartition des sommets d'une ligne polygonale régulière non fermée, Ann. Sci. École Norm. Sup. 73 (1956), 283-355. | Zbl 0072.03802

[003] [4] T. Komatsu, The fractional part of nθ + ϕ and Beatty sequences, J. Théor. Nombres Bordeaux 7 (1995), 387-406. | Zbl 0849.11027

[004] [5] T. Komatsu, On inhomogeneous continued fraction expansion and inhomogeneous Diophantine approximation, J. Number Theory 62 (1997), 192-212. | Zbl 0878.11029

[005] [6] K. Nishioka, I. Shiokawa and J. Tamura, Arithmetical properties of a certain power series, J. Number Theory. 42 (1992), 61-87. | Zbl 0770.11039

[006] [7] C. G. Pinner, personal communication.

[007] [8] T. van Ravenstein, The three gap theorem (Steinhaus conjecture), J. Austral. Math. Soc. Ser. A 45 (1988), 360-370. | Zbl 0663.10039

[008] [9] V. T. Sós, On the theory of Diophantine approximations, II, Acta Math. Acad. Sci. Hungar. 9 (1958), 229-241. | Zbl 0086.03902