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. Google Scholar

[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. Google Scholar

[3]

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

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

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

[10]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

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

[21]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[3]

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

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

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

[10]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

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

[21]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

Figure 1.  RCF-map and ECF-map
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