# Linear fractional transformations of continued fractions with bounded partial quotients

• Volume: 9, Issue: 2, page 267-279
• ISSN: 1246-7405

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## Abstract

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Let $\theta$ be a real number with continued fraction expansion$\theta =\left[{a}_{0},{a}_{1},{a}_{2},\cdots \right],$and let$M=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$be a matrix with integer entries and nonzero determinant. If $\theta$ has bounded partial quotients, then $\frac{a\theta +b}{c\theta +d}=\left[{a}_{0}^{*},{a}_{1}^{*},{a}_{2}^{*},\cdots \right]$ also has bounded partial quotients. More precisely, if ${a}_{j}\le K$ for all sufficiently large $j$, then ${a}_{j}^{*}\le |det\left(M\right)|\left(K+2\right)$ for all sufficiently large $j$. We also give a weaker bound valid for all ${a}_{j}^{*}$ with $j\ge 1$. The proofs use the homogeneous Diophantine approximation constant ${L}_{\infty }\left(\theta \right)={lim sup}_{q\to \infty }{\left(q∥{q}^{\theta }∥\right)}^{-1}$. We show that$\frac{1}{\left|det\left(M\right)\right|}{L}_{\infty }\left(\theta \right)\le {L}_{\infty }\left(\frac{a\theta +b}{c\theta +d}\right)\le \left|det\left(M\right)\right|{L}_{\infty }\left(\theta \right).$

## How to cite

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Lagarias, J. C., and Shallit, J. O.. "Linear fractional transformations of continued fractions with bounded partial quotients." Journal de théorie des nombres de Bordeaux 9.2 (1997): 267-279. <http://eudml.org/doc/248009>.

@article{Lagarias1997,
abstract = {Let $\theta$ be a real number with continued fraction expansion\begin\{equation*\}\theta = \left[ a\_0, a\_1, a\_2, \dots \right],\end\{equation*\}and let\begin\{equation*\}M = \begin\{bmatrix\}a & b \\ c & d \end\{bmatrix\}\end\{equation*\}be a matrix with integer entries and nonzero determinant. If $\theta$ has bounded partial quotients, then $\frac\{a \theta + b\}\{c \theta + d\} = \left[ a^\ast _0, a^\ast _1, a^\ast _2, \dots \right]$ also has bounded partial quotients. More precisely, if $a_j \le K$ for all sufficiently large $j$, then $a^\ast _j \le | \det (M)|(K + 2)$ for all sufficiently large $j$. We also give a weaker bound valid for all $a^\ast _j$ with $j \ge 1$. The proofs use the homogeneous Diophantine approximation constant $L_\infty \left( \theta \right) = \limsup _\{q \rightarrow \infty \} \left(q \left\Vert q^\theta \right\Vert \right)^\{-1\}$. We show that\begin\{equation*\} \frac\{1\}\{\left| \det (M) \right|\} L\_\infty ( \theta ) \le L\_\infty \left( \frac\{a \theta + b\}\{c \theta + d\} \right) \le \left| \det (M) \right| L\_\infty ( \theta ). \end\{equation*\}},
author = {Lagarias, J. C., Shallit, J. O.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {Lagrange spectrum; diophantine approximation; linear fractional transformation; continued fractions; bounded partial quotients},
language = {eng},
number = {2},
pages = {267-279},
publisher = {Université Bordeaux I},
title = {Linear fractional transformations of continued fractions with bounded partial quotients},
url = {http://eudml.org/doc/248009},
volume = {9},
year = {1997},
}

TY - JOUR
AU - Lagarias, J. C.
AU - Shallit, J. O.
TI - Linear fractional transformations of continued fractions with bounded partial quotients
JO - Journal de théorie des nombres de Bordeaux
PY - 1997
PB - Université Bordeaux I
VL - 9
IS - 2
SP - 267
EP - 279
AB - Let $\theta$ be a real number with continued fraction expansion\begin{equation*}\theta = \left[ a_0, a_1, a_2, \dots \right],\end{equation*}and let\begin{equation*}M = \begin{bmatrix}a & b \\ c & d \end{bmatrix}\end{equation*}be a matrix with integer entries and nonzero determinant. If $\theta$ has bounded partial quotients, then $\frac{a \theta + b}{c \theta + d} = \left[ a^\ast _0, a^\ast _1, a^\ast _2, \dots \right]$ also has bounded partial quotients. More precisely, if $a_j \le K$ for all sufficiently large $j$, then $a^\ast _j \le | \det (M)|(K + 2)$ for all sufficiently large $j$. We also give a weaker bound valid for all $a^\ast _j$ with $j \ge 1$. The proofs use the homogeneous Diophantine approximation constant $L_\infty \left( \theta \right) = \limsup _{q \rightarrow \infty } \left(q \left\Vert q^\theta \right\Vert \right)^{-1}$. We show that\begin{equation*} \frac{1}{\left| \det (M) \right|} L_\infty ( \theta ) \le L_\infty \left( \frac{a \theta + b}{c \theta + d} \right) \le \left| \det (M) \right| L_\infty ( \theta ). \end{equation*}
LA - eng
KW - Lagrange spectrum; diophantine approximation; linear fractional transformation; continued fractions; bounded partial quotients
UR - http://eudml.org/doc/248009
ER -

## References

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