A Liouville-like approach for the transcendence of some real numbers
type: Item
instance of: scholarly article
title: A Liouville-like approach for the transcendence of some real numbers
zbMATH Open document ID: 1046.11051
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2004-08-20T00:00:00Z
review text: The main result of this paper is the following: Theorem. Let \((u_n)_{n\geq0}\) be a strictly increasing sequence of positive integers. If there exists some \(\varepsilon>0\) such that \(u_{n+1}>(1+\varepsilon)u_n\) for infinitely many \(n\)'s, then the series \(\,f(z)=\sum_{k=0}^{+\infty} z^{u_k}\,\) takes transcendental values in each point \(z=1/\beta\) where \(\beta\) is a Pisot or Salem number. The proof involves a generalization of the Roth-Ridout theorem due to P. Corvaja.
Mathematics Subject Classification ID: 11J81
zbMATH DE Number: 2093920
zbMATH Keywords: transcendental numbers
zbMATH Keywords: Liouville numbers
author: Boris Adamczewski
reviewed by: Maurice Mignotte
MaRDI profile type: Publication
full work available at URL: j.crma.2004.02.002
OpenAlex ID: W1998510998
cites work: On Roth's theorem
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cites work: Rational approximations to algebraic numbers
cites work: Simple continued fractions for some irrational numbers
DOI: 10.1016/J.CRMA.2004.02.002
description: scientific article; zbMATH DE number 2093920