The efficiency of approximating real numbers by Lüroth expansion

• Volume: 63, Issue: 2, page 497-513
• ISSN: 0011-4642

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Abstract

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For any $x\in \left(0,1\right]$, let $x=\frac{1}{{d}_{1}}+\frac{1}{{d}_{1}\left({d}_{1}-1\right){d}_{2}}+\cdots +\frac{1}{{d}_{1}\left({d}_{1}-1\right)\cdots {d}_{n-1}\left({d}_{n-1}-1\right){d}_{n}}+\cdots$ be its Lüroth expansion. Denote by ${P}_{n}\left(x\right)/{Q}_{n}\left(x\right)$ the partial sum of the first $n$ terms in the above series and call it the $n$th convergent of $x$ in the Lüroth expansion. This paper is concerned with the efficiency of approximating real numbers by their convergents ${\left\{{P}_{n}\left(x\right)/{Q}_{n}\left(x\right)\right\}}_{n\ge 1}$ in the Lüroth expansion. It is shown that almost no points can have convergents as the optimal approximation for infinitely many times in the Lüroth expansion. Consequently, Hausdorff dimension is introduced to quantify the set of real numbers which can be well approximated by their convergents in the Lüroth expansion, namely the following Jarník-like set: $\left\{x\in \left(0,1\right]:|x-{P}_{n}\left(x\right)/{Q}_{n}\left(x\right)|<1/{Q}_{n}{\left(x\right)}^{\nu +1}\text{infinitely}\phantom{\rule{4.0pt}{0ex}}\text{often}\right\}$ for any $\nu \ge 1$.

How to cite

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Cao, Chunyun, Wu, Jun, and Zhang, Zhenliang. "The efficiency of approximating real numbers by Lüroth expansion." Czechoslovak Mathematical Journal 63.2 (2013): 497-513. <http://eudml.org/doc/260582>.

@article{Cao2013,
abstract = {For any $x\in (0,1]$, let $x=\frac\{1\}\{d\_1\}+\frac\{1\}\{d\_1(d\_1-1)d\_2\}+\dots +\frac\{1\}\{d\_1(d\_1-1) \dots d\_\{n-1\}(d\_\{n-1\}-1)d\_\{n\}\}+\dots$ be its Lüroth expansion. Denote by $\{P_n(x)\}/\{Q_n(x)\}$ the partial sum of the first $n$ terms in the above series and call it the $n$th convergent of $x$ in the Lüroth expansion. This paper is concerned with the efficiency of approximating real numbers by their convergents $\lbrace \{P_n(x)\}/\{Q_n(x)\}\rbrace _\{n\ge 1\}$ in the Lüroth expansion. It is shown that almost no points can have convergents as the optimal approximation for infinitely many times in the Lüroth expansion. Consequently, Hausdorff dimension is introduced to quantify the set of real numbers which can be well approximated by their convergents in the Lüroth expansion, namely the following Jarník-like set: $\lbrace x\in (0,1]\colon |x-\{P_n(x)\}/\{Q_n(x)\}|<\{1\}/\{Q_n(x)^\{\nu +1\}\} \text\{infinitely often\}\rbrace$ for any $\nu \ge 1$.},
author = {Cao, Chunyun, Wu, Jun, Zhang, Zhenliang},
journal = {Czechoslovak Mathematical Journal},
keywords = {Lüroth expansion; optimal approximation; Hausdorff dimension; Lüroth expansion; optimal approximation; Hausdorff dimension},
language = {eng},
number = {2},
pages = {497-513},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The efficiency of approximating real numbers by Lüroth expansion},
url = {http://eudml.org/doc/260582},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Cao, Chunyun
AU - Wu, Jun
AU - Zhang, Zhenliang
TI - The efficiency of approximating real numbers by Lüroth expansion
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 2
SP - 497
EP - 513
AB - For any $x\in (0,1]$, let $x=\frac{1}{d_1}+\frac{1}{d_1(d_1-1)d_2}+\dots +\frac{1}{d_1(d_1-1) \dots d_{n-1}(d_{n-1}-1)d_{n}}+\dots$ be its Lüroth expansion. Denote by ${P_n(x)}/{Q_n(x)}$ the partial sum of the first $n$ terms in the above series and call it the $n$th convergent of $x$ in the Lüroth expansion. This paper is concerned with the efficiency of approximating real numbers by their convergents $\lbrace {P_n(x)}/{Q_n(x)}\rbrace _{n\ge 1}$ in the Lüroth expansion. It is shown that almost no points can have convergents as the optimal approximation for infinitely many times in the Lüroth expansion. Consequently, Hausdorff dimension is introduced to quantify the set of real numbers which can be well approximated by their convergents in the Lüroth expansion, namely the following Jarník-like set: $\lbrace x\in (0,1]\colon |x-{P_n(x)}/{Q_n(x)}|<{1}/{Q_n(x)^{\nu +1}} \text{infinitely often}\rbrace$ for any $\nu \ge 1$.
LA - eng
KW - Lüroth expansion; optimal approximation; Hausdorff dimension; Lüroth expansion; optimal approximation; Hausdorff dimension
UR - http://eudml.org/doc/260582
ER -

References

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