Linear fractional transformations of continued fractions with bounded partial quotients

Lagarias, J. C.; Shallit, J. O.

Journal de théorie des nombres de Bordeaux, Tome 9 (1997) no. 2, pp. 267-279.

Soit $\theta$ un nombre réel de développement en fraction continue

$$\theta =\left[a_0,a_1,a_2,\cdots \right],$$

et soit

$$M=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$$

une matrice d’entiers tel que det $M\ne 0$. Si $\theta$ est à quotients partiels bornés, alors $\frac{a\theta +b}{c\theta +d}=\left[a_0^{*},a_1^{*},a_2^{*},\cdots \right]$ est aussi à quotients partiels bornés. Plus précisément, si $a_j\le K$ pour tout $j$ suffisamment grand, alors $a_j^{*}\le |det\left(M\right)|(K+2)$ pour tout $j$ suffisamment grand. Nous donnons aussi une borne plus faible qui est valable pour tout $a_j^{*}$ avec $j\ge 1$. Les démonstrations utilisent la constante d’approximation diophantienne homogène $L_{\infty }\left(\theta \right)={lim\; sup}_{q\to \infty }{\left(q∥q^{\theta }∥\right)}^{-1}$. Nous montrons que

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

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 $a_j^{*}\le |det\left(M\right)|(K+2)$ 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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