Dépendance linéaire de fonctions arithmétiques et presque arithmétiques (Linear dependence of arithmetical and almost arithmetical functions)
The author obtains by Schneider's method a very general transcendence theorem: Given an integer L, there exist two constants a and b with the following property. Let \(z_ n\) be a complex sequence and r(N) an upper bound for the \(| z_ n|\) (1\(\leq n\leq N)\). If \(f_ j(1\leq j\leq L)\) is an entire function such that \(f_ j(z_ n)\in {\mathbb{Z}}\) and log \(| f_ j(z)| \leq bN\) for \(| z| \leq a r(N)\), then the functions \(f_ j\) are \({\mathbb{Q}}\)-linearly dependent. This theorem contains for example Pólya's result (except for the value of the constant log 2): An entire function which maps \({\mathbb{N}}\) into \({\mathbb{Z}}\) and has a rate of growth smaller than exp(z log 2) is a polynomial. A similar proof gives a statement for functions such that \(f_ j(z_ n)\) is very near to an integer. This statement contains a particular case of a result by \textit{Ch. Pisot} [C. R. Acad. Sci., Paris 222, 1027-1028 (1946; Zbl 0060.215)]. Complete proofs can be found in ''Problèmes Diophantiens'' [Publ. Math. Univ. Paris VI, to appear].
scientific article; zbMATH DE number 3922769