Linear fractional transformations of continued fractions with bounded partial quotients

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 -