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On the Hausdorff dimension faithfulness of Oppenheim expansion

Volume 180 / 2017

Yu Sun, Zhengliang Zhang, Jia Liu Acta Arithmetica 180 (2017), 89-99 MSC: Primary 11K55; Secondary 28A80. DOI: 10.4064/aa8648-2-2017 Published online: 25 July 2017


We show that the family of cylinders generated by Oppenheim expansion is not faithful for Hausdorff dimension calculation on the unit interval. On the other hand, we prove that the family of all finite unions of consecutive cylinders of the same rank is faithful. Some special cases such as Lüroth expansion, Engel expansion and Sylvester expansion are included.


  • Yu SunFaculty of Science
    JiangSu University
    212013, Zhenjiang, P.R. China
  • Zhengliang ZhangSchool of Mathematical Sciences
    Henan Institute of Science and Technology
    453003, Xinxiang, P.R. China
  • Jia LiuInstitute of Statistics and Applied Mathematics
    Anhui University of Finance and Economics
    233030, Bengbu, P.R. China

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