# 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$\frac{1}{\left|det\left(M\right)\right|}{L}_{\infty }\left(\theta \right)\le {L}_{\infty }\left(\frac{a\theta +b}{c\theta +d}\right)\le \left|det\left(M\right)\right|{L}_{\infty }\left(\theta \right).$

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