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

[1] |
Doug Hensley. Continued fractions, Cantor sets, Hausdorff dimension, and transfer operators and their analytic extension. Discrete and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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
Other articles
by authors
[Back to Top]