On approximation by Lüroth series
;
Journal de Théorie des Nombres de Bordeaux, Volume 8 (1996) no. 2, pp. 331-346.

Let x]0,1] and p n /q n ,n1 be its sequence of Lüroth Series convergents. Define the approximation coefficients θ n =θ n (x) by q n x-p n ,n1. In [BBDK] the limiting distribution of the sequence (θ n ) n1 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 ) n1 and related sequences like (θ n +θ n+1 ) n1 . It turns out that for a.e. x the sequence (θ n ,θ n+1 ) n1 is distributed according to a continuous singular distribution function G. Furthermore we will see that two consecutive θ’s are positively correlated.

Pour x]0,1], on note p n /q n la suite des convergents de la série de Lüroth associée, et on définit par θ n =q n x-p n ,n1 ses coefficients d’approximation. Dans [BBDK], on détermine la fonction de répartition limite de la suite (θ n ), en utilisant l’extension naturelle du système ergodique sous-jacent au développement en série de Lüroth. Nous montrons ici que cela peut être fait sans cette considération. Plus précisément, nous démontrons que pour tout n, la répartition de θ n coïncide avec la répartition limite. On étudiera aussi la répartition pour presque tout x de la suite (θ n ,θ n+1 ) n1 , ainsi que celles issues de suites telles que (θ n +θ n+1 ) n1 . On obtiendra que pour presque tout x, la suite (θ n ,θ n+1 ) possède une fonction de répartition continue et singulière. On observera de plus que θ n et θ n+1 sont positivement corrélés.

@article{JTNB_1996__8_2_331_0,
     author = {Dajani, Karma and Kraaikamp, Cornelis},
     title = {On approximation by L\"uroth series},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux I},
     volume = {8},
     number = {2},
     year = {1996},
     pages = {331-346},
     zbl = {0870.11039},
     mrnumber = {1438473},
     language = {en},
     url={jtnb.centre-mersenne.org/item/JTNB_1996__8_2_331_0/}
}
Dajani, Karma; Kraaikamp, Cor. On approximation by Lüroth series. Journal de Théorie des Nombres de Bordeaux, Volume 8 (1996) no. 2, pp. 331-346. https://jtnb.centre-mersenne.org/item/JTNB_1996__8_2_331_0/

[BBDK] Barrionuevo, Jose, Robert M. Burton, Karma Dajani and Cor Kraaikamp - Ergodic Properties of Generalized Lüroth Series, Acta Arithm., LXXIV (4) (1996), 311-327. | MR 1378226 | Zbl 0848.11039

[DKS] Dajani, Karma, Cor Kraaikamp and Boris Solomyak - The natural extension of the β-transformation, Acta Math. Hungar., 73 (1-2) (1996), 97-109. | Zbl 0931.28014

[G] Galambos, J. - Representations of Real numbers by Infinite Series, Springer LNM 502, Springer-Verlag, Berlin, Heidelberg, New York, 1976. | MR 568141 | Zbl 0322.10002

[JK] Jager, H. and C. Kraaikamp - On the approximation by continued fractions, Indag. Math., 51 (1989), 289-307. | MR 1020023 | Zbl 0695.10029

[JdV] Jager, H. and C. De Vroedt - Lüroth series and their ergodic properties, Indag. Math. 31 (1968), 31-42. | Zbl 0167.32201

[L] Lüroth, J. - Ueber eine eindeutige Entwickelung von Zahlen in eine unendliche Reihe, Math. Annalen 21 (1883), 411-423. | JFM 15.0187.01 | MR 1510205

[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. | MR 115985 | Zbl 0090.03202

[T] Tucker, H.G. - A Graduate Course in Probability, Academic Press, New York, 1967. | MR 221541 | Zbl 0159.45702