Transcendence results on the generating functions of the characteristic functions of certain self-generating sets

Peter Bundschuh¹ ; Keijo Väänänen²

¹ Mathematisches Institut Universität zu Köln Weyertal 86-90 50931 Köln, Germany
² Department of Mathematical Sciences University of Oulu P.O. Box 3000 90014 Oulu, Finland

Acta Arithmetica, Tome 162 (2014) no. 3, pp. 273-288.

Résumé

This article continues two papers which recently appeared in this same journal. First, Dilcher and Stolarsky [140 (2009)] introduced two new power series, $F(z)$ and $G(z)$, related to the so-called Stern polynomials and having coefficients 0 and 1 only. Shortly later, Adamczewski [142 (2010)] proved, inter alia, that $G(\alpha ),G(\alpha ^4)$ are algebraically independent for any algebraic $\alpha $ with $0|\alpha |1$. \par Our first key result is that $F$ and $G$ have large blocks of consecutive zero coefficients. Then, a Roth-type argument shows that $F(a/b)$ and $G(a/b)$, for any $(a,b)\in \mathbb {Z}\times \mathbb {N}$ with $0|a|\sqrt {b}$, are transcendental but not U-numbers. Moreover, reasonably good upper bounds for the irrationality exponent of these numbers are obtained. Another main result for which an elementary (or poor men's) proof is presented concerns the algebraic independence of $F(z),F(z^4)$ over $\mathbb {C}(z)$ leading to the $F$-analogue of Adamczewski's above-mentioned theorem.

Détail

DOI : 10.4064/aa162-3-5

Keywords: article continues papers which recently appeared journal first dilcher stolarsky introduced power series related so called stern polynomials having coefficients only shortly later adamczewski proved inter alia alpha alpha algebraically independent algebraic alpha alpha par first key result have large blocks consecutive zero coefficients roth type argument shows mathbb times mathbb sqrt transcendental u numbers moreover reasonably upper bounds irrationality exponent these numbers obtained another main result which elementary poor mens proof presented concerns algebraic independence mathbb leading f analogue adamczewskis above mentioned theorem