Czechoslovak Mathematical Journal, first online, pp. 1-15


On the maximal run-length function in the Lüroth expansion

Yu Sun, Jian Xu

Received September 9, 2016.   First published January 18, 2018.

Abstract:  We obtain a metrical property on the asymptotic behaviour of the maximal run-length function in the Lüroth expansion. We also determine the Hausdorff dimension of a class of exceptional sets of points whose maximal run-length function has sub-linear growth rate.
Keywords:  Lüroth expansion; run-length function; Hausdorff dimension
Classification MSC:  11K55, 28A80
DOI:  10.21136/CMJ.2018.0474-16

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References:
[1] K. Dajani, C. Kraaikamp: Ergodic Theory of Numbers. Carus Mathematical Monographs 29, Mathematical Association of America, Washington (2002). MR 1917322 | Zbl 1033.11040
[2] P. Erdős, A. Rényi: On a new law of large numbers. J. Anal. Math. 23 (1970), 103-111. DOI 10.1007/BF02795493 | MR 0272026 | Zbl 0225.60015
[3] K. Falconer: Fractal Geometry: Mathematical Foundations and Applications. Wiley & Sons, Chichester (1990). MR 1102677 | Zbl 0689.28003
[4] J. Galambos: Representations of Real Numbers by Infinite Series. Lecture Notes in Mathematics 502, Springer, Berlin (1976). DOI 10.1007/BFb0081642 | MR 0568141 | Zbl 0322.10002
[5] J. E. Hutchinson: Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981), 713-747. DOI 10.1512/iumj.1981.30.30055 | MR 0625600 | Zbl 0598.28011
[6] M. Kesseböhmer, S. Munday, B. O. Stratmann: Infinite Ergodic Theory of Numbers. De Gruyter Graduate, De Gruyter, Berlin (2016). DOI 10.1515/9783110439427 | MR 3585883 | Zbl 1362.37003
[7] J. Li, M. Wu: On exceptional sets in Erdős-Rényi limit theorem. J. Math. Anal. Appl. 436 (2016), 355-365. DOI 10.1016/j.jmaa.2015.12.001 | MR 3440098 | Zbl 06536909
[8] J. Li, M. Wu: On exceptional sets in Erdős-Rényi limit theorem revisited. Monatsh. Math. 182 (2017), 865-875. DOI 10.1007/s00605-016-0977-y | MR 3624949 | Zbl 06704122
[9] J. Lüroth: Über eine eindeutige Entwickelung von Zahlen in eine unendliche Reihe. Math. Ann. 21 (1883), 411-423. (In German.) DOI 10.1007/BF01443883 | MR 1510205 | JFM 15.0187.01
[10] J.-H. Ma, S.-Y. Wen, Z.-Y. Wen: Egoroff's theorem and maximal run length. Monatsh. Math. 151 (2007), 287-292. DOI 10.1007/s00605-007-0455-7 | MR 2329089 | Zbl 1170.28001
[11] P. A. P. Moran: Additive functions of intervals and Hausdorff measure. Proc. Camb. Philos. Soc. 42 (1946), 15-23. DOI 10.1017/s0305004100022684 | MR 0014397 | Zbl 0063.04088
[12] P. Révész: Random Walk in Random and Non-random Environments. World Scientific, Hackensack (2005). DOI 10.1142/5847 | MR 2168855 | Zbl 1090.60001
[13] Y. Sun, J. Xu: A remark on exceptional sets in Erdős-Rényi limit theorem. Monatsh. Math. 184 (2017), 291-296. DOI 10.1007/s00605-016-0974-1 | MR 3696113 | Zbl 06788682
[14] B. W. Wang, J. Wu: On the maximal run-length function in continued fractions. Annales Univ. Sci. Budapest., Sect. Comp. 34 (2011), 247-268.
[15] R. Zou: Hausdorff dimension of the maximal run-length in dyadic expansion. Czech. Math. J. 61 (2011), 881-888. DOI 10.1007/s10587-011-0055-5 | MR 2886243 | Zbl 1249.11085

Affiliations:   Yu Sun, Faculty of Science, Jiangsu University, 301 Xuefu Rd, Jingkou, 212013 Zhenjiang, Jiangsu, P. R. China, e-mail: sunyu88sy@163.com; Jian Xu (corresponding author), School of Mathematics and Statistics, Huazhong University of Science and Technology, 1037 Luoyu Rd, Hongshan, 430074 Wuhan, Hubei, China, e-mail: arielxj@hotmail.com


 
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