Linear fractional transformations of continued fractions with bounded partial quotients

@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 -