# On approximation by Lüroth series

Journal de théorie des nombres de Bordeaux (1996)

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

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topDajani, 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

top- [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
- [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
- [G] Galambos, J. - Representations of Real numbers by Infinite Series, Springer LNM502, Springer-Verlag, Berlin, Heidelberg, New York, 1976. Zbl0322.10002MR568141
- [JK] Jager, H. and C. Kraaikamp - On the approximation by continued fractions, Indag. Math., 51 (1989), 289-307. Zbl0695.10029MR1020023
- [JdV] Jager, H. and C. de Vroedt - Lüroth series and their ergodic properties, Indag. Math.31 (1968), 31-42. Zbl0167.32201
- [L] Lüroth, J. - Ueber eine eindeutige Entwickelung von Zahlen in eine unendliche Reihe, Math. Annalen21 (1883), 411-423. MR1510205JFM15.0187.01
- [N] Nolte, Vincent N. - Some probabilistic results on continued fractions, Doktoraal scriptie Universiteit van Amsterdam, Amsterdam, August 1989.
- [Pe] Perron, O. - Irrationalzahlen, Walter de Gruyter & Co., Berlin, 1960. Zbl0090.03202MR115985
- [T] Tucker, H.G. - A Graduate Course in Probability, Academic Press, New York, 1967. Zbl0159.45702MR221541

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