Formes Lineaires d'Approximation et Irrationalite de Valeurs de $q$-Fonctions

Daniel Duverney

Abstract: Let $q$ be a rational integer, with $\vv q\geq 2$. Let $f$ be a complex function, analytic in $\vv x<R$, and satisfying a Poincaré-type equation of the form
$$ f(qx)=P(x)\, f(x) + Q(x), $$
where $P$ and $Q$ are rational fractions. We prove, under some conditions on $f$, that the set:
$$ \E(d)=\Bigl\{\alpha\in\Q\,/ R\vv q^{-1}\leq\vv\alpha <R\,;\ [\Q[f(\alpha)]:\Q]\leq d\Bigr\} $$ is finite, and give an upper bound (depending on $d$ and $f$) for the number of its elements.

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