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doi: 10.3934/dcds.2020389

Approximation properties of Lüroth expansions

Bo Tan 1, and  Qinglong Zhou 2,,

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  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). $
Keywords: Diophantine approximation, Lüroth expansion, Hausdorff dimension.
Mathematics Subject Classification: Primary: 11K55; Secondary: 28A80, 11J83.
Citation: Bo Tan, Qinglong Zhou. Approximation properties of Lüroth expansions. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020389
References:
[1]

J. Barrionuevo, R. M. Burton, K. Dajani and C. Kraaikamp, Ergodic properties of generalized Lüroth series, Acta Arith., 4 (1996), 311-327.  doi: 10.4064/aa-74-4-311-327.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

C. Y. Cao, J. 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.  Google Scholar

[6]

Y. H. Chen, Y. 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.  Google Scholar

[7]

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

[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.  Google Scholar

[9]

A. H. Fan, L. M. Liao, J. 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.  Google Scholar

[10]

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

[11]

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

[12]

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

[13]

M. Hussain, D. Kleinbock, N. Wadleigh and B. W. Wang, Hausdorff measure of sets of Dirichlet non-improvable numbers, Mathematika, 64 (2018), 502-518.  doi: 10.1112/S0025579318000074.  Google Scholar

[14]

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

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

B. W. Wang, J. 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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

J. Barrionuevo, R. M. Burton, K. Dajani and C. Kraaikamp, Ergodic properties of generalized Lüroth series, Acta Arith., 4 (1996), 311-327.  doi: 10.4064/aa-74-4-311-327.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

C. Y. Cao, J. 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.  Google Scholar

[6]

Y. H. Chen, Y. 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.  Google Scholar

[7]

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

[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.  Google Scholar

[9]

A. H. Fan, L. M. Liao, J. 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.  Google Scholar

[10]

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

[11]

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

[12]

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

[13]

M. Hussain, D. Kleinbock, N. Wadleigh and B. W. Wang, Hausdorff measure of sets of Dirichlet non-improvable numbers, Mathematika, 64 (2018), 502-518.  doi: 10.1112/S0025579318000074.  Google Scholar

[14]

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

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

B. W. Wang, J. 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.  Google Scholar

[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.  Google Scholar

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