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La complexité d’une suite infinie est définie comme la fonction qui compte le nombre de facteurs de longueur $k$ dans cette suite. Nous prouvons ici que la complexité des suites de Rudin-Shapiro généralisées (qui comptent les occurrences de certains facteurs dans les développements binaires d’entiers) est ultimement affine.

Every real number $x,0\<x\le 1$, has an essentially unique expansion as a Pierce series : $$x=\frac{1}{x^1}-\frac{1}{x^1{x}^2}+\frac{1}{x^1{x}^2{x}^3}-\cdots$$ where the $x_i$ form a strictly increasing sequence of positive integers. The expansion terminates if and only if $x$ is rational. Similarly, every positive real number $y$ has a unique expansion as an Engel series : $$y=\frac{1}{y^1}-\frac{1}{y^1{y}^2}+\frac{1}{y^1{y}^2{y}^3}+\cdots$$ where the $y_i$ form a (not necessarily strictly) increasing sequence of positive integers. If the expansion is infinite, we require that the sequence yi be not eventually...

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