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.

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     author = {Karma Dajani and Cor Kraaikamp},
     title = {On approximation by {L\"uroth} series},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {331--346},
     publisher = {Universit\'e Bordeaux I},
     volume = {8},
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     year = {1996},
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Karma Dajani; Cor Kraaikamp. On approximation by Lüroth series. Journal de Théorie des Nombres de Bordeaux, Volume 8 (1996) no. 2, pp. 331-346. doi : 10.5802/jtnb.172. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.172/

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