May 2018, 38(5): 2375-2393. doi: 10.3934/dcds.2018098

Hausdorff dimension of certain sets arising in Engel continued fractions

1. 

School of Mathematics, Sun Yat-sen University, Guangzhou, GD 510275, China

2. 

Department of Mathematics, South China University of Technology, Guangzhou, GD 510640, China

* Corresponding author: Lulu Fang

Received  July 2017 Revised  November 2017 Published  March 2018

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.
Citation: Lulu Fang, Min Wu. Hausdorff dimension of certain sets arising in Engel continued fractions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2375-2393. doi: 10.3934/dcds.2018098
References:
[1]

C.-Y. CaoB.-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.

[2]

T. Cusick, Hausdorff dimension of sets of continued fractions, Quart. J. Math. Oxford(2), 41 (1990), 277-286. doi: 10.1093/qmath/41.3.277.

[3]

K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, Ltd., Chichester, 1990.

[4]

A.-H. FanB.-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.

[5]

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.

[6]

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.

[7]

L.-L. FangM. 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.

[8]

D.-J. FengJ. WuJ.-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.

[9]

J. Galambos, Representations of Real Numbers by Infinite Series, Lecture Notes in Mathematics, Vol. 502. Springer-Verlag, Berlin-New York, 1976.

[10]

I. Good, The fractional dimensional theory of continued fractions, Proc. Cambridge Philos. Soc., 37 (1941), 199-228. doi: 10.1017/S030500410002171X.

[11]

P. HanusR. 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.

[12]

Y. HartonoC. 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.

[13]

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.

[14]

H. HuY.-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.

[15]

V. Jarník, Zur metrischen Theorie der diopahantischen Approximationen, Proc. Mat. Fyz., 36 (1928), 91-106.

[16]

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.

[17] A. Khintchine, Continued Fractions, The University of Chicago Press, Chicago-London, 1964.
[18]

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.

[19]

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.

[20]

T. Luczak, On the fractional dimension of sets of continued fractions, Mathematika, 44 (1997), 50-53. doi: 10.1112/S0025579300011955.

[21]

R. Mauldin and M. Urbański, Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc. (3), 73 (1996), 105-154.

[22]

L. Rempe-Gillen and M. Urbański, Non-autonomous conformal iterated function systems and Moran-set constructions, Trans. Amer. Math. Soc., 368 (2016), 1979-2017.

[23] F. Schweiger, Ergodic Theory of Fibred Systems and Metric Number Theory, Oxford University Press, New York, 1995.
[24]

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.

[25]

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.

[26]

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.

[27]

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.

show all references

References:
[1]

C.-Y. CaoB.-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.

[2]

T. Cusick, Hausdorff dimension of sets of continued fractions, Quart. J. Math. Oxford(2), 41 (1990), 277-286. doi: 10.1093/qmath/41.3.277.

[3]

K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, Ltd., Chichester, 1990.

[4]

A.-H. FanB.-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.

[5]

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.

[6]

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.

[7]

L.-L. FangM. 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.

[8]

D.-J. FengJ. WuJ.-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.

[9]

J. Galambos, Representations of Real Numbers by Infinite Series, Lecture Notes in Mathematics, Vol. 502. Springer-Verlag, Berlin-New York, 1976.

[10]

I. Good, The fractional dimensional theory of continued fractions, Proc. Cambridge Philos. Soc., 37 (1941), 199-228. doi: 10.1017/S030500410002171X.

[11]

P. HanusR. 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.

[12]

Y. HartonoC. 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.

[13]

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.

[14]

H. HuY.-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.

[15]

V. Jarník, Zur metrischen Theorie der diopahantischen Approximationen, Proc. Mat. Fyz., 36 (1928), 91-106.

[16]

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.

[17] A. Khintchine, Continued Fractions, The University of Chicago Press, Chicago-London, 1964.
[18]

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.

[19]

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.

[20]

T. Luczak, On the fractional dimension of sets of continued fractions, Mathematika, 44 (1997), 50-53. doi: 10.1112/S0025579300011955.

[21]

R. Mauldin and M. Urbański, Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc. (3), 73 (1996), 105-154.

[22]

L. Rempe-Gillen and M. Urbański, Non-autonomous conformal iterated function systems and Moran-set constructions, Trans. Amer. Math. Soc., 368 (2016), 1979-2017.

[23] F. Schweiger, Ergodic Theory of Fibred Systems and Metric Number Theory, Oxford University Press, New York, 1995.
[24]

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.

[25]

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.

[26]

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.

[27]

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.

Figure 1.  RCF-map and ECF-map
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