Hausdorff dimension of certain sets arising in Engel continued fractions

Lulu Fang; Min Wu

doi:10.3934/dcds.2018098

Previous Article
On Hausdorff dimension of the set of non-ergodic directions of two-genus double cover of tori
DCDS Home
This Issue
Next Article
Conormal derivative problems for stationary Stokes system in Sobolev spaces

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

Hausdorff dimension of certain sets arising in Engel continued fractions

Lulu Fang ^1,, and Min Wu ^2,

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.

Keywords: Engel continued fractions, growth rate of partial quotients, Hausdorff dimension.

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

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

References:

[1]	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. 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. 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. 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. 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. Google Scholar
[8]	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. 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. 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. Google Scholar
[12]	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. 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. 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. 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. Google Scholar
[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. Google Scholar
[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. 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. 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. 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. 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. 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. Google Scholar
[8]	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. 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. 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. Google Scholar
[12]	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. 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. 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. 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. Google Scholar
[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. Google Scholar
[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

Figure Options

Download full-size image

Download as PowerPoint slide

[1]	Doug Hensley. Continued fractions, Cantor sets, Hausdorff dimension, and transfer operators and their analytic extension. Discrete & Continuous Dynamical Systems, 2012, 32 (7) : 2417-2436. doi: 10.3934/dcds.2012.32.2417
[2]	Kanji Inui, Hikaru Okada, Hiroki Sumi. The Hausdorff dimension function of the family of conformal iterated function systems of generalized complex continued fractions. Discrete & Continuous Dynamical Systems, 2020, 40 (2) : 753-766. doi: 10.3934/dcds.2020060
[3]	Laura Luzzi, Stefano Marmi. On the entropy of Japanese continued fractions. Discrete & Continuous Dynamical Systems, 2008, 20 (3) : 673-711. doi: 10.3934/dcds.2008.20.673
[4]	Pierre Arnoux, Thomas A. Schmidt. Commensurable continued fractions. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4389-4418. doi: 10.3934/dcds.2014.34.4389
[5]	Claudio Bonanno, Carlo Carminati, Stefano Isola, Giulio Tiozzo. Dynamics of continued fractions and kneading sequences of unimodal maps. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1313-1332. doi: 10.3934/dcds.2013.33.1313
[6]	Élise Janvresse, Benoît Rittaud, Thierry de la Rue. Dynamics of $\lambda$-continued fractions and $\beta$-shifts. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1477-1498. doi: 10.3934/dcds.2013.33.1477
[7]	Marc Kessböhmer, Bernd O. Stratmann. On the asymptotic behaviour of the Lebesgue measure of sum-level sets for continued fractions. Discrete & Continuous Dynamical Systems, 2012, 32 (7) : 2437-2451. doi: 10.3934/dcds.2012.32.2437
[8]	Hiroki Sumi, Mariusz Urbański. Bowen parameter and Hausdorff dimension for expanding rational semigroups. Discrete & Continuous Dynamical Systems, 2012, 32 (7) : 2591-2606. doi: 10.3934/dcds.2012.32.2591
[9]	Sara Munday. On Hausdorff dimension and cusp excursions for Fuchsian groups. Discrete & Continuous Dynamical Systems, 2012, 32 (7) : 2503-2520. doi: 10.3934/dcds.2012.32.2503
[10]	Shmuel Friedland, Gunter Ochs. Hausdorff dimension, strong hyperbolicity and complex dynamics. Discrete & Continuous Dynamical Systems, 1998, 4 (3) : 405-430. doi: 10.3934/dcds.1998.4.405
[11]	Luis Barreira and Jorg Schmeling. Invariant sets with zero measure and full Hausdorff dimension. Electronic Research Announcements, 1997, 3: 114-118.
[12]	Jon Chaika. Hausdorff dimension for ergodic measures of interval exchange transformations. Journal of Modern Dynamics, 2008, 2 (3) : 457-464. doi: 10.3934/jmd.2008.2.457
[13]	Krzysztof Barański, Michał Wardal. On the Hausdorff dimension of the Sierpiński Julia sets. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3293-3313. doi: 10.3934/dcds.2015.35.3293
[14]	Stéphane Sabourau. Growth of quotients of groups acting by isometries on Gromov-hyperbolic spaces. Journal of Modern Dynamics, 2013, 7 (2) : 269-290. doi: 10.3934/jmd.2013.7.269
[15]	Federico Rodriguez Hertz, María Alejandra Rodriguez Hertz, Raúl Ures. Partial hyperbolicity and ergodicity in dimension three. Journal of Modern Dynamics, 2008, 2 (2) : 187-208. doi: 10.3934/jmd.2008.2.187
[16]	Thomas Jordan, Mark Pollicott. The Hausdorff dimension of measures for iterated function systems which contract on average. Discrete & Continuous Dynamical Systems, 2008, 22 (1&2) : 235-246. doi: 10.3934/dcds.2008.22.235
[17]	Vanderlei Horita, Marcelo Viana. Hausdorff dimension for non-hyperbolic repellers II: DA diffeomorphisms. Discrete & Continuous Dynamical Systems, 2005, 13 (5) : 1125-1152. doi: 10.3934/dcds.2005.13.1125
[18]	Krzysztof Barański. Hausdorff dimension of self-affine limit sets with an invariant direction. Discrete & Continuous Dynamical Systems, 2008, 21 (4) : 1015-1023. doi: 10.3934/dcds.2008.21.1015
[19]	Carlos Matheus, Jacob Palis. An estimate on the Hausdorff dimension of stable sets of non-uniformly hyperbolic horseshoes. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 431-448. doi: 10.3934/dcds.2018020
[20]	Aline Cerqueira, Carlos Matheus, Carlos Gustavo Moreira. Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra. Journal of Modern Dynamics, 2018, 12: 151-174. doi: 10.3934/jmd.2018006

2020 Impact Factor: 1.392

Tools

Metrics

Tools

Article outline

Figures and Tables

2020 Impact Factor: 1.392

Tools

Figures and Tables

2020 Impact Factor: 1.392

American Institute of Mathematical Sciences

Hausdorff dimension of certain sets arising in Engel continued fractions

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Hausdorff dimension of certain sets arising in Engel continued fractions

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors