Article
| Title: | Run-length function of the Bolyai-Rényi expansion of real numbers (English) |
| Author: | Li, Rao |
| Author: | Lü, Fan |
| Author: | Zhou, Li |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 74 |
| Issue: | 1 |
| Year: | 2024 |
| Pages: | 319-335 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | By iterating the Bolyai-Rényi transformation $T(x)=(x+1)^{2} \pmod 1$, almost every real number $x\in [0,1)$ can be expanded as a continued radical expression $$ x=-1+\sqrt {x_{1}+\sqrt {x_{2}+\cdots +\sqrt {x_{n}+\cdots }}} $$ with digits $x_{n}\in \{0,1,2\}$ for all $n\in \mathbb {N}$. For any real number $x\in [0,1)$ and digit $i\in \{0,1,2\}$, let $r_{n}(x,i)$ be the maximal length of consecutive $i$'s in the first $n$ digits of the Bolyai-Rényi expansion of $x$. We study the asymptotic behavior of the run-length function $r_{n}(x,i)$. We prove that for any digit $i\in \{0,1,2\}$, the Lebesgue measure of the set $$ D(i)=\Bigl \{x\in [0,1)\colon \lim _{n\rightarrow \infty } \frac {r_n(x,i)}{\log n}=\frac {1}{\log \theta _{i}} \Bigr \} $$ is $1$, where $\theta _{i}=1+\sqrt {4i+1}$. We also obtain that the level set $$ E_{\alpha }(i)=\Bigl \{x\in [0,1)\colon \lim _{n\rightarrow \infty } \frac {r_n(x,i)}{\log n}=\alpha \Bigr \} $$ is of full Hausdorff dimension for any $0\leq \alpha \leq \infty $. (English) |
| Keyword: | run-length function |
| Keyword: | Bolyai-Rényi expansion |
| Keyword: | Lebesgue measure |
| Keyword: | Hausdorff dimension |
| MSC: | 11K55 |
| MSC: | 28A80 |
| DOI: | 10.21136/CMJ.2023.0351-23 |
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| Date available: | 2024-03-13T10:12:00Z |
| Last updated: | 2024-03-18 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152283 |
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| Reference: | [1] Erdős, P., Rényi, A.: On a new law of large numbers.J. Anal. Math. 23 (1970), 103-111. Zbl 0225.60015, MR 0272026, 10.1007/BF02795493 |
| Reference: | [2] Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications.John Wiley & Sons, Chichester (2014). Zbl 1285.28011, MR 3236784, 10.1002/0470013850 |
| Reference: | [3] Jenkinson, O., Pollicott, M.: Ergodic properties of the Bolyai-Rényi expansion.Indag. Math., New Ser. 11 (2000), 399-418. Zbl 0977.11032, MR 1813480, 10.1016/S0019-3577(00)80006-3 |
| Reference: | [4] Ma, J.-H., Wen, S.-Y., Wen, Z.-Y.: Egoroff's theorem and maximal run length.Monatsh. Math. 151 (2007), 287-292. Zbl 1170.28001, MR 2329089, 10.1007/s00605-007-0455-7 |
| Reference: | [5] Philipp, W.: Some metrical theorems in number theory.Pac. J. Math. 20 (1967), 109-127. Zbl 0144.04201, MR 0205930, 10.2140/pjm.1967.20.109 |
| Reference: | [6] Rényi, A.: Representations for real numbers and their ergodic properties.Acta Math. Acad. Sci. Hung. 8 (1957), 477-493. Zbl 0079.08901, MR 0097374, 10.1007/BF02020331 |
| Reference: | [7] Song, T., Zhou, Q.: On the longest block function in continued fractions.Bull. Aust. Math. Soc. 102 (2020), 196-206. Zbl 1464.11080, MR 4138819, 10.1017/S0004972720000076 |
| Reference: | [8] 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: | [9] Tong, X., Yu, Y., Zhao, Y.: On the maximal length of consecutive zero digits of $\beta$-expansions.Int. J. Number Theory 12 (2016), 625-633. Zbl 1337.11053, MR 3477410, 10.1142/S1793042116500408 |
| Reference: | [10] Wang, B.-W., Wu, J.: On the maximal run-length function in continued fractions.Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Comput. 34 (2011), 247-268. |
| Reference: | [11] Zou, R.: Hausdorff dimension of the maximal run-length in dyadic expansion.Czech. Math. J. 61 (2011), 881-888. Zbl 1249.11085, MR 2886243, 10.1007/s10587-011-0055-5 |
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