no. 5
Théorie « élémentaire » des nombres
The relative growth rate for the digits in Lüroth expansions
Tan, Xiaoyan1 ; Zhang, Zhenliang2 
1 School of Mathematical Sciences, Henan Institute of Science and Technology, Xinxiang, 453003, P. R. China
2 School of Mathematical Sciences, Chongqing Normal University, Chongqing, 401131, P. R.China
Comptes Rendus. Mathématique, Tome 358 (2020) no. 5, pp. 557-562.

In this note, the rate of growth of digits in the Lüroth expansion of an irrational number is studied relative to the rate of approximation of the number by its convergents. The Hausdorff dimension of exceptional sets of points with a given relative growth rate is established.

Reçu le :
Révisé le :
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Publié le :
DOI : 10.5802/crmath.71
Classification : 11K55, 28A80
Mots clés : Lüroth expansion, Hausdorff dimension, relative growth rate
Tan, Xiaoyan 1 ; Zhang, Zhenliang 2

1 School of Mathematical Sciences, Henan Institute of Science and Technology, Xinxiang, 453003, P. R. China
2 School of Mathematical Sciences, Chongqing Normal University, Chongqing, 401131, P. R.China 
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Tan, Xiaoyan; Zhang, Zhenliang. The relative growth rate for the digits in Lüroth expansions. Comptes Rendus. Mathématique, Tome 358 (2020) no. 5, pp. 557-562. doi : 10.5802/crmath.71. http://www.numdam.org/articles/10.5802/crmath.71/

[1] Barreira, Luis; Iommi, Godofredo Frequency of digits in the Lüroth expansion, J. Number Theory, Volume 129 (2009) no. 6, pp. 1479-1490 | DOI | Zbl

[2] Cao, Chunyun; Wu, Jun; Zhang, Zhenliang The efficiency of approximating real numbers by Lüroth expansion, Czech. Math. J., Volume 63 (2013) no. 2, pp. 497-513 | Zbl

[3] Falconer, Kenneth J. Fractal geometry. Mathematical foundations and applications, John Wiley & Sons, 2014 | Zbl

[4] Galambos, Janos Reprentations of real numbers by infinite series, Lecture Notes in Mathematics, 502, Springer, 1976 | Zbl

[5] Hass, Andrew The relative growth rate for partial quotients, New York J. Math., Volume 14 (2008), pp. 139-143 | MR | Zbl

[6] Li, Jinjun; Wu, Min; Yang, Xianfeng On the longest block in Lüroth expansion, J. Math. Anal. Appl., Volume 457 (2018) no. 1, pp. 522-532 | Zbl

[7] Liao, Lingmin; Rams, Michal Big Birkhoff sums in d-decaying Gauss like iterated function systems (2019) (https://arxiv.org/abs/1905.02547)

[8] Lüroth, Jacob Ueber eine eindeutige entwickelung von zahlen in eine unendliche reihe, Math. Ann., Volume 21 (1883) no. 2, pp. 411-423 | DOI | MR

[9] Shen, Luming; Fang, Kui The fractional dimensional theory in Lüroth expansion, Czech. Math. J., Volume 61 (2011) no. 3, pp. 795-807 | DOI | Zbl

[10] Shen, Luming; Yu, Yiying; Zhou, Yuxin A note on the largest digits in Lüroth expansion, Int. J. Number Theory, Volume 10 (2014) no. 4, pp. 1015-1023 | DOI | Zbl

[11] Song, Kunkun; Fang, Lulu; Ma, Jihua Level sets of partial maximal digits for Lüroth expansion, Int. J. Number Theory, Volume 13 (2017) no. 10, pp. 2777-2790 | DOI | Zbl

[12] Sun, Yu; Wu, Jun A dimensional result in continued fractions, Int. J. Number Theory, Volume 10 (2014) no. 4, pp. 849-857 | DOI | MR | Zbl

[13] Tan, Bo; Zhou, Qinglong The relative growth rate for partial quotients in continued fractions, J. Math. Anal. Appl., Volume 478 (2019) no. 1, pp. 229-235 | DOI | MR | Zbl

[14] Tan, Xiaoyan; He, Kaijan A note on the relative growth rate of the maximal digit in Lüroth expansions, Fractals (2020) (“online first”, https://doi.org/10.1142/S0218348X20501169) | DOI

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