On inhomogeneous diophantine approximation with some quasi-periodic expressions, II

Takao Komatsu

Journal de théorie des nombres de Bordeaux (1999)

  • Volume: 11, Issue: 2, page 331-344
  • ISSN: 1246-7405

Abstract

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We consider the values concerning ℳ ( θ , φ ) = lim inf | q | → ∞ | q | | | q θ - φ | | where the continued fraction expansion of θ has a quasi-periodic form. In particular, we treat the cases so that each quasi-periodic form includes no constant. Furthermore, we give some general conditions satisfying ℳ ( θ , φ ) = 0 .

How to cite

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Komatsu, Takao. "On inhomogeneous diophantine approximation with some quasi-periodic expressions, II." Journal de théorie des nombres de Bordeaux 11.2 (1999): 331-344. <http://eudml.org/doc/248343>.

@article{Komatsu1999,
abstract = {We consider the values concerning\begin\{equation*\} \mathcal \{M\}(\theta , \phi ) = \liminf \_\{|q| \rightarrow \infty \} |q|||q^\theta - \phi || \end\{equation*\}where the continued fraction expansion of $\theta $ has a quasi-periodic form. In particular, we treat the cases so that each quasi-periodic form includes no constant. Furthermore, we give some general conditions satisfying $\mathcal \{M\}(\theta , \phi ) = 0$.},
author = {Komatsu, Takao},
journal = {Journal de théorie des nombres de Bordeaux},
language = {eng},
number = {2},
pages = {331-344},
publisher = {Université Bordeaux I},
title = {On inhomogeneous diophantine approximation with some quasi-periodic expressions, II},
url = {http://eudml.org/doc/248343},
volume = {11},
year = {1999},
}

TY - JOUR
AU - Komatsu, Takao
TI - On inhomogeneous diophantine approximation with some quasi-periodic expressions, II
JO - Journal de théorie des nombres de Bordeaux
PY - 1999
PB - Université Bordeaux I
VL - 11
IS - 2
SP - 331
EP - 344
AB - We consider the values concerning\begin{equation*} \mathcal {M}(\theta , \phi ) = \liminf _{|q| \rightarrow \infty } |q|||q^\theta - \phi || \end{equation*}where the continued fraction expansion of $\theta $ has a quasi-periodic form. In particular, we treat the cases so that each quasi-periodic form includes no constant. Furthermore, we give some general conditions satisfying $\mathcal {M}(\theta , \phi ) = 0$.
LA - eng
UR - http://eudml.org/doc/248343
ER -

References

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  1. [1] J.W.S. Cassels, Über limx→+∞x|ϑx + α — y|. Math. Ann.127 (1954), 288-304. Zbl0055.04401
  2. [2] T W. Cusick, A. M. Rockett and P. Szüsz, On inhomogeneous Diophantine approximation. J. Number Theory48 (1994), 259-283. Zbl0820.11042MR1293862
  3. [3] C.S. DavisOn some simple continued fractions connected with e. J. London Math. Soc.20 (1945), 194-198. Zbl0060.16111MR17394
  4. [4] R. DescombesSur la répartition des sommets d'une ligne polygonale régulière non fermée. Ann. Sci. École Norm Sup.73 (1956), 283-355. Zbl0072.03802MR86844
  5. [5] T. Komatsu On inhomogeneouscontinued fraction expansion and inhomogeneous Diophantine approximation. J. Number Theory62 (1997), 192-212. Zbl0878.11029MR1430009
  6. [6] T. KomatsuOn inhomogeneous Diophantine approximation and the Nishioka-Shiokawa-Tamura algorithm. Acta Arith.86 (1998), 305-324. Zbl0930.11049MR1659089
  7. [7] T. KomatsuOn inhomogeneous Diophantine approximation with some quasi-periodic expressions. Acta Math. Hung.85 (1999), 303-322. Zbl0949.11035MR1724097
  8. [8] K.R. Matthews and R.F.C. WaltersSome properties of the continued fraction expansion of (m/n)e1/q. Proc. Cambridge Philos. Soc.67 (1970), 67-74. Zbl0188.10703MR252889
  9. [9] K. Nishioka, I. Shiokawa and J. TamuraArithmetical properties of a certain power series. J. Number Theory42 (1992), 61-87. Zbl0770.11039MR1176421
  10. [10] O. PerronDie Lehre von den Kettenbrüchen. Chelsea reprint of 1929 edition. Zbl0041.18206
  11. [11] V.T. SósOn the theory of Diophantine approximations, II. Acta Math. Acad. Sci. Hung.9 (1958), 229-241. Zbl0086.03902MR95164

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