Linear fractional transformations of continued fractions with bounded partial quotients
Let \(\theta\) be a real number with continued fraction expansion \(\theta=[a_0,a_1,a_2,\dots]\), and let \(M=\left[\begin{smallmatrix} a&b\\ c&d \end{smallmatrix} \right]\) be an integer matrix with \(\det(M)\neq 0\). Set \(K(\theta)=\sup_{i\geq 1}a_i\) and \(K_\infty\limsup_{i\geq 1}a_i\). Then \[ {1\over| \det(M)| }K_\infty(\theta)-2\leq K_\infty \Biggl({a\theta+b\over c\theta+d}\Biggr) \leq| \det(M)| (K_\infty(\theta)+2) \] and \[ K\Biggl( {a\theta+b\over c\theta+d} \Biggr)\leq| \det(M)| (K(\theta)+2)+| c(c\theta+d)| . \] The proofs use the type \(L(\theta)=\sup_{q\geq 1}(q\| q\theta\|)^{-1}\) and the Lagrange constant \[ L_\infty(\theta)= \limsup_{q\geq 1}(q\| q\theta\|)^{-1} \] for an irrational \(\theta\). The former follows from the facts \(L_{\infty}({a\theta+b\over c\theta+d})=L_{\infty}(\theta)\), and \(\sup(L_{\infty}({a\theta+b\over c\theta+d}))=m L_{\infty}(\theta)\) and \(\inf(L_{\infty}({a\theta+b\over c\theta+d}))\geq{1\over m}L_{\infty}(\theta)\), where \(M\) takes all the matrices such that \(| \det(M)| =m\). The latter follows from the result \[ L\Biggl( {a\theta+b\over c\theta+d} \Biggr)\leq| \det(M)| L(\theta)+| c(c\theta+d)| . \] The authors also show as one of the remarks that \[ {1\over| \det(M)| }L_\infty(\theta)\leq L_\infty \Biggl({a\theta+b\over c\theta+d}\Biggr) \leq| \det(M)| L_\infty(\theta). \]
scientific article; zbMATH DE number 1164087
scientific article