Article
| Title: | Hausdorff dimension of some exceptional sets in Lüroth expansions (English) |
| Author: | Wang, Ao |
| Author: | Zhang, Xinyun |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 75 |
| Issue: | 2 |
| Year: | 2025 |
| Pages: | 461-484 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | We study the metrical theory of the growth rate of digits in Lüroth expansions. More precisely, for $ x\in ( 0,1 ]$, let $[ d_1( x ) ,d_2 ( x) ,\cdots ]$ denote the Lüroth expansion of $x$. We completely determine the Hausdorff dimension of the sets $$ \begin{aligned} E_{\sup } ( \psi ) = & \biggl \{ x\in ( 0,1 ] \colon \limsup _{n\rightarrow \infty } \frac {\log d_n ( x)}{\psi ( n )}=1 \biggr \} ,\\ E ( \psi ) = & \biggl \{ x\in ( 0,1 ] \colon \lim _{n\rightarrow \infty } \frac {\log d_n ( x)}{\psi ( n )}=1 \biggr \} \end{aligned} $$ and $$ E_{\inf } (\psi ) =\biggl \{ x\in ( 0,1 ] \colon \liminf _{n\rightarrow \infty } \frac {\log d_n ( x )}{\psi ( n)}=1 \biggr \} , $$ where $ \psi \colon \mathbb {N} \rightarrow \mathbb {R} ^+ $ is an arbitrary function satisfying $ \psi ( n ) \rightarrow \infty $ as $n\rightarrow \infty $. (English) |
| Keyword: | Lüroth expansion |
| Keyword: | exceptional set |
| Keyword: | Borel-Bernstein theorem |
| Keyword: | Hausdorff dimension |
| MSC: | 11K55 |
| MSC: | 28A80 |
| DOI: | 10.21136/CMJ.2025.0109-24 |
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| Date available: | 2025-05-20T11:44:26Z |
| Last updated: | 2025-05-26 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152953 |
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| Reference: | [1] Arroyo, A., Robert, G. González: Hausdorff dimension of sets of numbers with large Lüroth elements.Integers 21 (2021), Article ID a71, 20 pages. Zbl 1469.11250, MR 4284749 |
| Reference: | [2] Barreira, L., Iommi, G.: Frequency of digits in the Lüroth expansion.J. Number Theory 129 (2009), 1479-1490. Zbl 1188.37024, MR 2521488, 10.1016/j.jnt.2008.06.002 |
| Reference: | [3] Cao, C., Wu, J., Zhang, Z.: The efficiency of approximating real numbers by Lüroth expansion.Czech. Math. J. 63 (2013), 497-513. Zbl 1289.11045, MR 3073974, 10.1007/s10587-013-0033-1 |
| Reference: | [4] Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications.John Wiley & Sons, Chichester (2014). Zbl 1285.28011, MR 3236784, 10.1002/0470013850 |
| Reference: | [5] Fan, A., Liao, L., Ma, J., Wang, B.: Dimension of Besicovitch-Eggleston sets in countable symbolic space.Nonlinearity 23 (2010), 1185-1197. Zbl 1247.11104, MR 2630097, 10.1088/0951-7715/23/5/009 |
| Reference: | [6] Fang, L., Ma, J., Song, K.: Some exceptional sets of Borel-Bernstein theorem in continued fractions.Ramanujan J. 56 (2021), 891-909. Zbl 1481.11078, MR 4341100, 10.1007/s11139-020-00320-8 |
| Reference: | [7] Galambos, J.: Representations of Real Numbers by Infinite Series.Lecture Notes in Mathematics 502. Springer, Berlin (1976). Zbl 0322.10002, MR 568141, 10.1007/BFb0081642 |
| Reference: | [8] Lüroth, J.: Ueber eine eindeutige Entwickelung von Zahlen in eine unendliche Reihe.Math. Ann. 21 (1883), 411-423 German \99999JFM99999 15.0187.01. MR 1510205, 10.1007/BF01443883 |
| Reference: | [9] Šalát, T.: Zur metrischen Theorie der Lürothschen Entwicklungen der reellen Zahlen.Czech. Math. J. 18 (1968), 489-522 German. Zbl 0162.34703, MR 229605, 10.21136/CMJ.1968.100848 |
| Reference: | [10] Shen, L.: Hausdorff dimension of the set concerning with Borel-Bernstein theory in Lüroth expansions.J. Korean Math. Soc. 54 (2017), 1301-1316. Zbl 1368.41002, MR 3668870, 10.4134/JKMS.j160501 |
| Reference: | [11] Shen, L., Fang, K.: The fractional dimensional theory in Lüroth expansion.Czech. Math. J. 61 (2011), 795-807. Zbl 1249.11084, MR 2853093, 10.1007/s10587-011-0028-8 |
| Reference: | [12] Sun, Y., Xu, J.: On the maximal run-length function in the Lüroth expansion.Czech. Math. J. 68 (2018), 277-291. Zbl 1458.11125, MR 3783599, 10.21136/CMJ.2018.0474-16 |
| Reference: | [13] Tan, B., Zhou, Q.: Dimension theory of the product of partial quotients in Lüroth expansions.Int. J. Number Theory 17 (2021), 1139-1154. Zbl 1469.11256, MR 4270877, 10.1142/S1793042121500287 |
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