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Hausdorff dimension of certain sets arising in Engel continued fractions

  • * Corresponding author: Lulu Fang

    * Corresponding author: Lulu Fang 
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  • In the present paper, we are concerned with the Hausdorff dimension of certain sets arising in Engel continued fractions. In particular, the Hausdorff dimension of sets

    $\big\{x ∈ [0,1): b_n(x) ≥ \phi (n)~i.m.~n ∈ \mathbb{N}\big\}\ \ \text{and}\ \ \big\{x ∈ [0,1): b_n(x) ≥ \phi(n),\ \forall n ≥ 1\big\}$

    are completely determined, where $i.m.$ means infinitely many, $\{b_n(x)\}_{n ≥ 1}$ is the sequence of partial quotients of the Engel continued fraction expansion of $x$ and $\phi$ is a positive function defined on natural numbers.

    Mathematics Subject Classification: Primary: 11K50, 37E05; Secondary: 28A80.

    Citation:

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  •   C.-Y. Cao , B.-W. Wang  and  J. Wu , The growth speed of digits in infinite iterated function systems, Studia Math., 217 (2013) , 139-158.  doi: 10.4064/sm217-2-3.
      T. Cusick , Hausdorff dimension of sets of continued fractions, Quart. J. Math. Oxford, 41 (1990) , 277-286.  doi: 10.1093/qmath/41.3.277.
      K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, Ltd., Chichester, 1990.
      A.-H. Fan , B.-W. Wang  and  J. Wu , Arithmetic and metric properties of Oppenheim continued fraction expansions, J. Number Theory, 127 (2007) , 64-82.  doi: 10.1016/j.jnt.2006.12.016.
      L.-L. Fang , Large and moderate deviations for modified Engel continued fractions, Statist. Probab. Lett., 98 (2015) , 98-106.  doi: 10.1016/j.spl.2014.12.015.
      L. -L. Fang and M. Wu, A note on Rényi's "record" problem and Engel's series, to appear in Proceedings of the Edinburgh Mathematical Society.
      L.-L. Fang , M. Wu  and  L. Shang , Large and moderate deviation principles for Engel continued fraction, J. Theoret. Probab., (2015) , 1-25.  doi: 10.1007/s10959-016-0715-3.
      D.-J. Feng , J. Wu , J.-C. Liang  and  S. Tseng , Appendix to the paper by T. Luczak-a simple proof of the lower bound: "On the fractional dimension of sets of continued fractions", Mathematika, 44 (1997) , 54-55.  doi: 10.1112/S0025579300011967.
      J. Galambos, Representations of Real Numbers by Infinite Series, Lecture Notes in Mathematics, Vol. 502. Springer-Verlag, Berlin-New York, 1976.
      I. Good , The fractional dimensional theory of continued fractions, Proc. Cambridge Philos. Soc., 37 (1941) , 199-228.  doi: 10.1017/S030500410002171X.
      P. Hanus , R. Mauldin  and  M. Urbański , Thermodynamic formalism and multifractal analysis of conformal infinite iterated function systems, Acta Math. Hungar., 96 (2002) , 27-98.  doi: 10.1023/A:1015613628175.
      Y. Hartono , C. Kraaikamp  and  F. Schweiger , Algebraic and ergodic properties of a new continued fraction algorithm with non-decreasing partial quotients, J. Théor. Nombres Bordeaux, 14 (2002) , 497-516.  doi: 10.5802/jtnb.371.
      K. Hirst , Continued fractions with sequences of partial quotients, Proc. Amer. Math. Soc., 38 (1973) , 221-227.  doi: 10.1090/S0002-9939-1973-0311581-4.
      H. Hu , Y.-L. Yu  and  Y.-F. Zhao , A note on approximation efficiency and partial quotients of Engel continued fractions, Int. J. Number Theory, 13 (2017) , 2433-2443.  doi: 10.1142/S1793042117501329.
      V. Jarník , Zur metrischen Theorie der diopahantischen Approximationen, Proc. Mat. Fyz., 36 (1928) , 91-106. 
      T. Jordan  and  M. Rams , Increasing digit subsystems of infinite iterated function systems, Proc. Amer. Math. Soc., 140 (2012) , 1267-1279.  doi: 10.1090/S0002-9939-2011-10969-9.
      A. KhintchineContinued Fractions, The University of Chicago Press, Chicago-London, 1964. 
      C. Kraaikamp  and  J. Wu , On a new continued fraction expansion with non-decreasing partial quotients, Monatsh. Math., 143 (2004) , 285-298.  doi: 10.1007/s00605-004-0246-3.
      L.-M. Liao  and  M. Rams , Subexponentially increasing sums of partial quotients in continued fraction expansions, Math. Proc. Cambridge Philos. Soc., 160 (2016) , 401-412.  doi: 10.1017/S0305004115000742.
      T. Luczak , On the fractional dimension of sets of continued fractions, Mathematika, 44 (1997) , 50-53.  doi: 10.1112/S0025579300011955.
      R. Mauldin  and  M. Urbański , Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc. (3), 73 (1996) , 105-154. 
      L. Rempe-Gillen  and  M. Urbański , Non-autonomous conformal iterated function systems and Moran-set constructions, Trans. Amer. Math. Soc., 368 (2016) , 1979-2017. 
      F. SchweigerErgodic Theory of Fibred Systems and Metric Number Theory, Oxford University Press, New York, 1995. 
      B.-W. Wang  and  J. Wu , A problem of Hirst on continued fractions with sequences of partial quotients, Bull. Lond. Math. Soc., 40 (2008) , 18-22.  doi: 10.1112/blms/bdm103.
      B.-W. Wang  and  J. Wu , Hausdorff dimension of certain sets arising in continued fraction expansions, Adv. Math., 218 (2008) , 1319-1339.  doi: 10.1016/j.aim.2008.03.006.
      Z.-L. Zhang  and  C.-Y. Cao , On points with positive density of the digit sequence in infinite iterated function systems, J. Aust. Math. Soc., 102 (2017) , 435-443.  doi: 10.1017/S1446788716000288.
      T. Zhong  and  L. Tang , The sets of different continued fractions with the same partial quotients, Int. J. Number Theory, 9 (2013) , 1855-1863.  doi: 10.1142/S1793042113500619.
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