On inhomogeneous Diophantine approximation and the Nishioka-Shiokawa-Tamura algorithm

Takao Komatsu

Acta Arithmetica (1998)

  • Volume: 86, Issue: 4, page 305-324
  • ISSN: 0065-1036

Abstract

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We obtain the values concerning ( θ , ϕ ) = l i m i n f | q | → ∞ | q | ‖ q θ - 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,...].

How to cite

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Takao Komatsu. "On inhomogeneous Diophantine approximation and the Nishioka-Shiokawa-Tamura algorithm." Acta Arithmetica 86.4 (1998): 305-324. <http://eudml.org/doc/207199>.

@article{TakaoKomatsu1998,
abstract = {We obtain the values concerning $ (θ,ϕ) = lim inf_\{|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,...].},
author = {Takao Komatsu},
journal = {Acta Arithmetica},
keywords = {Nishioka-Shiokawa-Tamura algorithm; inhomogeneous approximation constant; continued fraction expansion},
language = {eng},
number = {4},
pages = {305-324},
title = {On inhomogeneous Diophantine approximation and the Nishioka-Shiokawa-Tamura algorithm},
url = {http://eudml.org/doc/207199},
volume = {86},
year = {1998},
}

TY - JOUR
AU - Takao Komatsu
TI - On inhomogeneous Diophantine approximation and the Nishioka-Shiokawa-Tamura algorithm
JO - Acta Arithmetica
PY - 1998
VL - 86
IS - 4
SP - 305
EP - 324
AB - We obtain the values concerning $ (θ,ϕ) = lim inf_{|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,...].
LA - eng
KW - Nishioka-Shiokawa-Tamura algorithm; inhomogeneous approximation constant; continued fraction expansion
UR - http://eudml.org/doc/207199
ER -

References

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  1. [1] J. W. S. Cassels, Über l ̲ i m x → + ∞ x | ϑ x + α - y | , Math. Ann. 127 (1954), 288-304. 
  2. [2] T. W. Cusick, A. M. Rockett and P. Szüsz, On inhomogeneous Diophantine approximation, J. Number Theory 48 (1994), 259-283. Zbl0820.11042
  3. [3] R. Descombes, Sur la répartition des sommets d'une ligne polygonale régulière non fermée, Ann. Sci. École Norm. Sup. 73 (1956), 283-355. Zbl0072.03802
  4. [4] T. Komatsu, The fractional part of nθ + ϕ and Beatty sequences, J. Théor. Nombres Bordeaux 7 (1995), 387-406. Zbl0849.11027
  5. [5] T. Komatsu, On inhomogeneous continued fraction expansion and inhomogeneous Diophantine approximation, J. Number Theory 62 (1997), 192-212. Zbl0878.11029
  6. [6] K. Nishioka, I. Shiokawa and J. Tamura, Arithmetical properties of a certain power series, J. Number Theory. 42 (1992), 61-87. Zbl0770.11039
  7. [7] C. G. Pinner, personal communication. 
  8. [8] T. van Ravenstein, The three gap theorem (Steinhaus conjecture), J. Austral. Math. Soc. Ser. A 45 (1988), 360-370. Zbl0663.10039
  9. [9] V. T. Sós, On the theory of Diophantine approximations, II, Acta Math. Acad. Sci. Hungar. 9 (1958), 229-241. Zbl0086.03902

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