Run-length function of the Bolyai-Rényi expansion of real numbers

Rao Li; Fan Lü; Li Zhou

Czechoslovak Mathematical Journal (2024)

  • Volume: 74, Issue: 1, page 319-335
  • ISSN: 0011-4642

Abstract

top
By iterating the Bolyai-Rényi transformation T ( x ) = ( x + 1 ) 2 ( mod 1 ) , almost every real number x [ 0 , 1 ) can be expanded as a continued radical expression x = - 1 + x 1 + x 2 + + x n + with digits x n { 0 , 1 , 2 } for all n . For any real number x [ 0 , 1 ) and digit i { 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 { 0 , 1 , 2 } , the Lebesgue measure of the set D ( i ) = x [ 0 , 1 ) : lim n r n ( x , i ) log n = 1 log θ i is 1 , where θ i = 1 + 4 i + 1 . We also obtain that the level set E α ( i ) = x [ 0 , 1 ) : lim n r n ( x , i ) log n = α is of full Hausdorff dimension for any 0 α .

How to cite

top

Li, Rao, Lü, Fan, and Zhou, Li. "Run-length function of the Bolyai-Rényi expansion of real numbers." Czechoslovak Mathematical Journal 74.1 (2024): 319-335. <http://eudml.org/doc/299246>.

@article{Li2024,
abstract = {By iterating the Bolyai-Rényi transformation $T(x)=(x+1)^\{2\} \hspace\{4.44443pt\}(\@mod \; 1)$, 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 \lbrace 0,1,2\rbrace $ for all $n\in \mathbb \{N\}$. For any real number $x\in [0,1)$ and digit $i\in \lbrace 0,1,2\rbrace $, 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 \lbrace 0,1,2\rbrace $, the Lebesgue measure of the set \[ D(i)=\Bigl \lbrace x\in [0,1)\colon \lim \_\{n\rightarrow \infty \} \frac\{r\_n(x,i)\}\{\log n\}=\frac\{1\}\{\log \theta \_\{i\}\} \Bigr \rbrace \] is $1$, where $\theta _\{i\}=1+\sqrt\{4i+1\}$. We also obtain that the level set \[ E\_\{\alpha \}(i)=\Bigl \lbrace x\in [0,1)\colon \lim \_\{n\rightarrow \infty \} \frac\{r\_n(x,i)\}\{\log n\}=\alpha \Bigr \rbrace \] is of full Hausdorff dimension for any $0\le \alpha \le \infty $.},
author = {Li, Rao, Lü, Fan, Zhou, Li},
journal = {Czechoslovak Mathematical Journal},
keywords = {run-length function; Bolyai-Rényi expansion; Lebesgue measure; Hausdorff dimension},
language = {eng},
number = {1},
pages = {319-335},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Run-length function of the Bolyai-Rényi expansion of real numbers},
url = {http://eudml.org/doc/299246},
volume = {74},
year = {2024},
}

TY - JOUR
AU - Li, Rao
AU - Lü, Fan
AU - Zhou, Li
TI - Run-length function of the Bolyai-Rényi expansion of real numbers
JO - Czechoslovak Mathematical Journal
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 74
IS - 1
SP - 319
EP - 335
AB - By iterating the Bolyai-Rényi transformation $T(x)=(x+1)^{2} \hspace{4.44443pt}(\@mod \; 1)$, 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 \lbrace 0,1,2\rbrace $ for all $n\in \mathbb {N}$. For any real number $x\in [0,1)$ and digit $i\in \lbrace 0,1,2\rbrace $, 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 \lbrace 0,1,2\rbrace $, the Lebesgue measure of the set \[ D(i)=\Bigl \lbrace x\in [0,1)\colon \lim _{n\rightarrow \infty } \frac{r_n(x,i)}{\log n}=\frac{1}{\log \theta _{i}} \Bigr \rbrace \] is $1$, where $\theta _{i}=1+\sqrt{4i+1}$. We also obtain that the level set \[ E_{\alpha }(i)=\Bigl \lbrace x\in [0,1)\colon \lim _{n\rightarrow \infty } \frac{r_n(x,i)}{\log n}=\alpha \Bigr \rbrace \] is of full Hausdorff dimension for any $0\le \alpha \le \infty $.
LA - eng
KW - run-length function; Bolyai-Rényi expansion; Lebesgue measure; Hausdorff dimension
UR - http://eudml.org/doc/299246
ER -

References

top
  1. Erdős, P., Rényi, A., 10.1007/BF02795493, J. Anal. Math. 23 (1970), 103-111. (1970) Zbl0225.60015MR0272026DOI10.1007/BF02795493
  2. Falconer, K., 10.1002/0470013850, John Wiley & Sons, Chichester (2014). (2014) Zbl1285.28011MR3236784DOI10.1002/0470013850
  3. Jenkinson, O., Pollicott, M., 10.1016/S0019-3577(00)80006-3, Indag. Math., New Ser. 11 (2000), 399-418. (2000) Zbl0977.11032MR1813480DOI10.1016/S0019-3577(00)80006-3
  4. Ma, J.-H., Wen, S.-Y., Wen, Z.-Y., 10.1007/s00605-007-0455-7, Monatsh. Math. 151 (2007), 287-292. (2007) Zbl1170.28001MR2329089DOI10.1007/s00605-007-0455-7
  5. Philipp, W., 10.2140/pjm.1967.20.109, Pac. J. Math. 20 (1967), 109-127. (1967) Zbl0144.04201MR0205930DOI10.2140/pjm.1967.20.109
  6. Rényi, A., 10.1007/BF02020331, Acta Math. Acad. Sci. Hung. 8 (1957), 477-493. (1957) Zbl0079.08901MR0097374DOI10.1007/BF02020331
  7. Song, T., Zhou, Q., 10.1017/S0004972720000076, Bull. Aust. Math. Soc. 102 (2020), 196-206. (2020) Zbl1464.11080MR4138819DOI10.1017/S0004972720000076
  8. Sun, Y., Xu, J., 10.21136/CMJ.2018.0474-16, Czech. Math. J. 68 (2018), 277-291. (2018) Zbl1458.11125MR3783599DOI10.21136/CMJ.2018.0474-16
  9. Tong, X., Yu, Y., Zhao, Y., 10.1142/S1793042116500408, Int. J. Number Theory 12 (2016), 625-633. (2016) Zbl1337.11053MR3477410DOI10.1142/S1793042116500408
  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. (2011) 
  11. Zou, R., 10.1007/s10587-011-0055-5, Czech. Math. J. 61 (2011), 881-888. (2011) Zbl1249.11085MR2886243DOI10.1007/s10587-011-0055-5

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.