# The efficiency of approximating real numbers by Lüroth expansion

Chunyun Cao; Jun Wu; Zhenliang Zhang

Czechoslovak Mathematical Journal (2013)

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

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topCao, 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 -

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