June  2021, 41(6): 2873-2890. doi: 10.3934/dcds.2020389

Approximation properties of Lüroth expansions

1. 

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

2. 

School of Science, Wuhan University of Technology, Wuhan 430074, China

* Corresponding author: Qinglong Zhou

Received  April 2020 Revised  September 2020 Published  June 2021 Early access  December 2020

For
$ x\in[0,1), $
let
$ [d_{1}(x),d_{2}(x),\ldots] $
be its Lüroth expansion and
$ \big\{\frac{p_{n}(x)}{q_{n}(x)}, n\geq 1\big\} $
be the sequence of convergents of
$ x. $
In this paper, we study the Jarník-like set of real numbers which can be well approximated by infinitely many of their convergents in the Lüroth expansion
$ \colon $
$ W(\psi) = \{x\in[0,1)\colon |xq_{n}(x)-p_{n}(x)|<\psi(n) \text{ for infinitely many } n\in \mathbb{N}\}, $
where
$ \psi\colon \mathbb{R}\to (0,\frac{1}{2}] $
is a positive function. We completely determine the Hausdorff dimension of
$ W(\psi). $
Citation: Bo Tan, Qinglong Zhou. Approximation properties of Lüroth expansions. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2873-2890. doi: 10.3934/dcds.2020389
References:
[1]

J. BarrionuevoR. M. BurtonK. Dajani and C. Kraaikamp, Ergodic properties of generalized Lüroth series, Acta Arith., 4 (1996), 311-327.  doi: 10.4064/aa-74-4-311-327.

[2]

J. Barral and S. Seuret, A localized Jarník-Besicovitch theorem, Adv. Math., 4 (2011), 3191-3215.  doi: 10.1016/j.aim.2010.10.011.

[3]

L. Barreira and G. Iommi, Frequency of digits in the Lüroth expansion, J. Number Theory, 6 (2009), 1479-1490.  doi: 10.1016/j.jnt.2008.06.002.

[4]

V. Beresnevich and S. Velani, A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures, Ann. of Math., 3 (2006), 971-992.  doi: 10.4007/annals.2006.164.971.

[5]

C. Y. CaoJ. Wu and Z. L. Zhang, The efficiency of approximating real numbers by Lüroth expansion, Czechoslovak Math. J., 2 (2013), 497-513.  doi: 10.1007/s10587-013-0033-1.

[6]

Y. H. ChenY. Sun and X. J. Zhao, A localized uniformly Jarník set in continued fractions, Acta Arith., 167 (2015), 267-280.  doi: 10.4064/aa167-3-5.

[7]

K. Dajani and C. Kraaikamp, On approximation by Lüroth series, J. Théor. Nombres Bordeaux, 8 (1996), 331-346.  doi: 10.5802/jtnb.172.

[8]

R. J. Duffin and A. C. Schaeffer, Khintchine's problem in metric Diophantine approximation, Duke Math. J., 8 (1941), 243-255.  doi: 10.1215/S0012-7094-41-00818-9.

[9]

A. H. FanL. M. LiaoJ. H. Ma and B. W. Wang, Dimension of Besicovitch-Eggleston sets in countable symbolic space, Nonlinearity, 23 (2010), 1185-1197.  doi: 10.1088/0951-7715/23/5/009.

[10]

K. J. Falconer, Fractal Geometry, Mathematical Foundations and Applications, 3$^{rd}$ edition, John Wiley & Sons, Ltd., Chichester, 2014.

[11]

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

[12]

J. Galambos, Representations of real numbers by infinite series, Lecture Notes in Mathematics, vol. 502, Springer-Verlag, Berlin-New York, 1976.

[13]

M. HussainD. KleinbockN. Wadleigh and B. W. Wang, Hausdorff measure of sets of Dirichlet non-improvable numbers, Mathematika, 64 (2018), 502-518.  doi: 10.1112/S0025579318000074.

[14]

V. Jarník, Zur Theorie der diophantischen Approximationen, Monatsh. Math. Phys., 39 (1932), 403-438.  doi: 10.1007/BF01699082.

[15]

A. Ya. Khintchine, Einige Sätzeber Kettenbrche, mit Anwendungen auf die Theorie der Diophantischen Approximationen, Math. Ann., 92 (1924), 115-125.  doi: 10.1007/BF01448437.

[16]

A. Ya. Khintchine, Continued Fractions, P. Noordhoff, Ltd., Groningen, 1963.

[17]

D. Koukoulopoulos and J. Maynard, On the Duffin-Schaeffer conjecture, Ann. of Math., 192 (2020), 251-307.  doi: 10.4007/annals.2020.192.1.5.

[18]

J. Lüroth, Ueber eine eindeutige Entwickelung von Zahlen in eine unendliche Reihe, Math. Ann., 21 (1883), 411-423.  doi: 10.1007/BF01443883.

[19]

R. D. Mauldin and M. Urbański, Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc., 73 (1996), 105-154.  doi: 10.1112/plms/s3-73.1.105.

[20]

L. M. Shen, Hausdorff dimension of the set concerning with Borel-Bernstein theory in Lüroth expansions, J. Korean Math. Soc., 54 (2017), 1301-1316.  doi: 10.4134/JKMS.j160501.

[21]

W. M. Schmidt, Diophantine Approximation, Lecture Notes in Mathematics, vol. 785, Springer, Berlin, 1980. doi: 3-540-09762-7.

[22]

B. Tan and Q. L. Zhou, The relative growth rate for partial quotients in continued fractions, J. Math. Anal. Appl., 478 (2019), 229-235.  doi: 10.1016/j.jmaa.2019.05.029.

[23]

B. Tan and Q.L. Zhou, Dimension theory of the product of partial quotients in Lüroth expansions, Int. J. Number Theory, (2020). doi: 10.1142/S1793042121500287.

[24]

B. W. Wang and J. Wu, Hausdorff dimension of certain sets arising in continued fraction expansions, Adv. Math., 5 (2008), 1319-1339.  doi: 10.1016/j.aim.2008.03.006.

[25]

B. W. WangJ. Wu and J. Xu, A generalization of the Jarník-Besicovitch theorem by continued fractions, Ergodic Theory Dynam. Systems, 36 (2016), 1278-1306.  doi: 10.1017/etds.2014.98.

[26]

S. K. Wang and J. Xu, On the Lebesgue measure of sum-level sets for Lüroth expansion, J. Math. Anal. Appl., 374 (2011), 197-200.  doi: 10.1016/j.jmaa.2010.08.047.

show all references

References:
[1]

J. BarrionuevoR. M. BurtonK. Dajani and C. Kraaikamp, Ergodic properties of generalized Lüroth series, Acta Arith., 4 (1996), 311-327.  doi: 10.4064/aa-74-4-311-327.

[2]

J. Barral and S. Seuret, A localized Jarník-Besicovitch theorem, Adv. Math., 4 (2011), 3191-3215.  doi: 10.1016/j.aim.2010.10.011.

[3]

L. Barreira and G. Iommi, Frequency of digits in the Lüroth expansion, J. Number Theory, 6 (2009), 1479-1490.  doi: 10.1016/j.jnt.2008.06.002.

[4]

V. Beresnevich and S. Velani, A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures, Ann. of Math., 3 (2006), 971-992.  doi: 10.4007/annals.2006.164.971.

[5]

C. Y. CaoJ. Wu and Z. L. Zhang, The efficiency of approximating real numbers by Lüroth expansion, Czechoslovak Math. J., 2 (2013), 497-513.  doi: 10.1007/s10587-013-0033-1.

[6]

Y. H. ChenY. Sun and X. J. Zhao, A localized uniformly Jarník set in continued fractions, Acta Arith., 167 (2015), 267-280.  doi: 10.4064/aa167-3-5.

[7]

K. Dajani and C. Kraaikamp, On approximation by Lüroth series, J. Théor. Nombres Bordeaux, 8 (1996), 331-346.  doi: 10.5802/jtnb.172.

[8]

R. J. Duffin and A. C. Schaeffer, Khintchine's problem in metric Diophantine approximation, Duke Math. J., 8 (1941), 243-255.  doi: 10.1215/S0012-7094-41-00818-9.

[9]

A. H. FanL. M. LiaoJ. H. Ma and B. W. Wang, Dimension of Besicovitch-Eggleston sets in countable symbolic space, Nonlinearity, 23 (2010), 1185-1197.  doi: 10.1088/0951-7715/23/5/009.

[10]

K. J. Falconer, Fractal Geometry, Mathematical Foundations and Applications, 3$^{rd}$ edition, John Wiley & Sons, Ltd., Chichester, 2014.

[11]

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

[12]

J. Galambos, Representations of real numbers by infinite series, Lecture Notes in Mathematics, vol. 502, Springer-Verlag, Berlin-New York, 1976.

[13]

M. HussainD. KleinbockN. Wadleigh and B. W. Wang, Hausdorff measure of sets of Dirichlet non-improvable numbers, Mathematika, 64 (2018), 502-518.  doi: 10.1112/S0025579318000074.

[14]

V. Jarník, Zur Theorie der diophantischen Approximationen, Monatsh. Math. Phys., 39 (1932), 403-438.  doi: 10.1007/BF01699082.

[15]

A. Ya. Khintchine, Einige Sätzeber Kettenbrche, mit Anwendungen auf die Theorie der Diophantischen Approximationen, Math. Ann., 92 (1924), 115-125.  doi: 10.1007/BF01448437.

[16]

A. Ya. Khintchine, Continued Fractions, P. Noordhoff, Ltd., Groningen, 1963.

[17]

D. Koukoulopoulos and J. Maynard, On the Duffin-Schaeffer conjecture, Ann. of Math., 192 (2020), 251-307.  doi: 10.4007/annals.2020.192.1.5.

[18]

J. Lüroth, Ueber eine eindeutige Entwickelung von Zahlen in eine unendliche Reihe, Math. Ann., 21 (1883), 411-423.  doi: 10.1007/BF01443883.

[19]

R. D. Mauldin and M. Urbański, Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc., 73 (1996), 105-154.  doi: 10.1112/plms/s3-73.1.105.

[20]

L. M. Shen, Hausdorff dimension of the set concerning with Borel-Bernstein theory in Lüroth expansions, J. Korean Math. Soc., 54 (2017), 1301-1316.  doi: 10.4134/JKMS.j160501.

[21]

W. M. Schmidt, Diophantine Approximation, Lecture Notes in Mathematics, vol. 785, Springer, Berlin, 1980. doi: 3-540-09762-7.

[22]

B. Tan and Q. L. Zhou, The relative growth rate for partial quotients in continued fractions, J. Math. Anal. Appl., 478 (2019), 229-235.  doi: 10.1016/j.jmaa.2019.05.029.

[23]

B. Tan and Q.L. Zhou, Dimension theory of the product of partial quotients in Lüroth expansions, Int. J. Number Theory, (2020). doi: 10.1142/S1793042121500287.

[24]

B. W. Wang and J. Wu, Hausdorff dimension of certain sets arising in continued fraction expansions, Adv. Math., 5 (2008), 1319-1339.  doi: 10.1016/j.aim.2008.03.006.

[25]

B. W. WangJ. Wu and J. Xu, A generalization of the Jarník-Besicovitch theorem by continued fractions, Ergodic Theory Dynam. Systems, 36 (2016), 1278-1306.  doi: 10.1017/etds.2014.98.

[26]

S. K. Wang and J. Xu, On the Lebesgue measure of sum-level sets for Lüroth expansion, J. Math. Anal. Appl., 374 (2011), 197-200.  doi: 10.1016/j.jmaa.2010.08.047.

[1]

Shrikrishna G. Dani. Simultaneous diophantine approximation with quadratic and linear forms. Journal of Modern Dynamics, 2008, 2 (1) : 129-138. doi: 10.3934/jmd.2008.2.129

[2]

Dmitry Kleinbock, Barak Weiss. Dirichlet's theorem on diophantine approximation and homogeneous flows. Journal of Modern Dynamics, 2008, 2 (1) : 43-62. doi: 10.3934/jmd.2008.2.43

[3]

Chao Ma, Baowei Wang, Jun Wu. Diophantine approximation of the orbits in topological dynamical systems. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2455-2471. doi: 10.3934/dcds.2019104

[4]

Sanghoon Kwon, Seonhee Lim. Equidistribution with an error rate and Diophantine approximation over a local field of positive characteristic. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 169-186. doi: 10.3934/dcds.2018008

[5]

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

[6]

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

[7]

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

[8]

Luis Barreira and Jorg Schmeling. Invariant sets with zero measure and full Hausdorff dimension. Electronic Research Announcements, 1997, 3: 114-118.

[9]

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

[10]

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

[11]

Thierry Paul, Mario Pulvirenti. Asymptotic expansion of the mean-field approximation. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 1891-1921. doi: 10.3934/dcds.2019080

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

Cristina Lizana, Leonardo Mora. Lower bounds for the Hausdorff dimension of the geometric Lorenz attractor: The homoclinic case. Discrete and Continuous Dynamical Systems, 2008, 22 (3) : 699-709. doi: 10.3934/dcds.2008.22.699

[20]

Davit Karagulyan. Hausdorff dimension of a class of three-interval exchange maps. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1257-1281. doi: 10.3934/dcds.2020077

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (287)
  • HTML views (148)
  • Cited by (0)

Other articles
by authors

[Back to Top]