On approximation by Lüroth series

Karma Dajani; Cor Kraaikamp

Journal de théorie des nombres de Bordeaux (1996)

  • Volume: 8, Issue: 2, page 331-346
  • ISSN: 1246-7405

Abstract

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Let x ∈ ] 0 , 1 ] and p n / q n , n ≥ 1 be its sequence of Lüroth Series convergents. Define the approximation coefficients θ n = θ n ( x ) by q n x - p n , n ≥ 1 . In [BBDK] the limiting distribution of the sequence ( θ n ) n ≥ 1 was obtained for a.e. x using the natural extension of the ergodic system underlying the Lüroth Series expansion. Here we show that this can be done without the natural extension. In fact we will prove that for each n , θ n is already distributed according to the limiting distribution. Using the natural extension we will study the distribution for a.e. x of the sequence ( θ n , θ n + 1 ) n ≥ 1 and related sequences like ( θ n + θ n + 1 ) n ≥ 1 . It turns out that for a.e. x the sequence ( θ n , θ n + 1 ) n ≥ 1 is distributed according to a continuous singular distribution function G. Furthermore we will see that two consecutive θ ’s are positively correlated.

How to cite

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Dajani, Karma, and Kraaikamp, Cor. "On approximation by Lüroth series." Journal de théorie des nombres de Bordeaux 8.2 (1996): 331-346. <http://eudml.org/doc/247818>.

@article{Dajani1996,
abstract = {Let $x \in ] 0, 1 ]$ and $p_n/q_n, n \ge 1$ be its sequence of Lüroth Series convergents. Define the approximation coefficients $\theta _n = \theta _n( x)$ by $q_nx - p_n, n \ge 1$. In [BBDK] the limiting distribution of the sequence $(\theta _n)_\{n \ge 1\}$ was obtained for a.e. $x$ using the natural extension of the ergodic system underlying the Lüroth Series expansion. Here we show that this can be done without the natural extension. In fact we will prove that for each $n, \theta _ n$ is already distributed according to the limiting distribution. Using the natural extension we will study the distribution for a.e. $x$ of the sequence $(\theta _n, \theta _\{n +1\})_\{n \ge 1\}$ and related sequences like $(\theta _n + \theta _\{n +1\})_\{n \ge 1\}$. It turns out that for a.e. $x$ the sequence $(\theta _n, \theta _\{n +1\})_\{n \ge 1\}$ is distributed according to a continuous singular distribution function G. Furthermore we will see that two consecutive $\theta $’s are positively correlated.},
author = {Dajani, Karma, Kraaikamp, Cor},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {Lüroth expansion; metric results; correlation coefficient},
language = {eng},
number = {2},
pages = {331-346},
publisher = {Université Bordeaux I},
title = {On approximation by Lüroth series},
url = {http://eudml.org/doc/247818},
volume = {8},
year = {1996},
}

TY - JOUR
AU - Dajani, Karma
AU - Kraaikamp, Cor
TI - On approximation by Lüroth series
JO - Journal de théorie des nombres de Bordeaux
PY - 1996
PB - Université Bordeaux I
VL - 8
IS - 2
SP - 331
EP - 346
AB - Let $x \in ] 0, 1 ]$ and $p_n/q_n, n \ge 1$ be its sequence of Lüroth Series convergents. Define the approximation coefficients $\theta _n = \theta _n( x)$ by $q_nx - p_n, n \ge 1$. In [BBDK] the limiting distribution of the sequence $(\theta _n)_{n \ge 1}$ was obtained for a.e. $x$ using the natural extension of the ergodic system underlying the Lüroth Series expansion. Here we show that this can be done without the natural extension. In fact we will prove that for each $n, \theta _ n$ is already distributed according to the limiting distribution. Using the natural extension we will study the distribution for a.e. $x$ of the sequence $(\theta _n, \theta _{n +1})_{n \ge 1}$ and related sequences like $(\theta _n + \theta _{n +1})_{n \ge 1}$. It turns out that for a.e. $x$ the sequence $(\theta _n, \theta _{n +1})_{n \ge 1}$ is distributed according to a continuous singular distribution function G. Furthermore we will see that two consecutive $\theta $’s are positively correlated.
LA - eng
KW - Lüroth expansion; metric results; correlation coefficient
UR - http://eudml.org/doc/247818
ER -

References

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  1. [BBDK] Barrionuevo, Jose, Robert M. Burton, Karma Dajani and Cor Kraaikamp - Ergodic Properties of Generalized Lüroth Series, Acta Arithm., LXXIV (4) (1996), 311-327. Zbl0848.11039MR1378226
  2. [DKS] Dajani, Karma, Cor Kraaikamp and Boris Solomyak - The natural extension of the β-transformation, Acta Math. Hungar., 73 (1-2) (1996), 97-109. Zbl0931.28014
  3. [G] Galambos, J. - Representations of Real numbers by Infinite Series, Springer LNM502, Springer-Verlag, Berlin, Heidelberg, New York, 1976. Zbl0322.10002MR568141
  4. [JK] Jager, H. and C. Kraaikamp - On the approximation by continued fractions, Indag. Math., 51 (1989), 289-307. Zbl0695.10029MR1020023
  5. [JdV] Jager, H. and C. de Vroedt - Lüroth series and their ergodic properties, Indag. Math.31 (1968), 31-42. Zbl0167.32201
  6. [L] Lüroth, J. - Ueber eine eindeutige Entwickelung von Zahlen in eine unendliche Reihe, Math. Annalen21 (1883), 411-423. MR1510205JFM15.0187.01
  7. [N] Nolte, Vincent N. - Some probabilistic results on continued fractions, Doktoraal scriptie Universiteit van Amsterdam, Amsterdam, August 1989. 
  8. [Pe] Perron, O. - Irrationalzahlen, Walter de Gruyter & Co., Berlin, 1960. Zbl0090.03202MR115985
  9. [T] Tucker, H.G. - A Graduate Course in Probability, Academic Press, New York, 1967. Zbl0159.45702MR221541

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