Continued fractional measure of irrationality
type: Item
instance of: scholarly article
title: Continued fractional measure of irrationality
zbMATH Open document ID: 1221.11157
DOI: 10.1215/0023608X-2009-002
author: Jaroslav Hančl
author: Simona Pulcerová
author: Tapani Matala-Aho
published in: Kyoto Journal of Mathematics
publication date: 2010-06-07T00:00:00Z
review text: Es sei \([a_1,a_2,\dots]\) der reguläre Kettenbruch von \(a\in\mathbb{R}\setminus\mathbb{Q}, a>1\) und \(p_n/q_n\) sein \(n\)-ter Näherungsbruch; \(I(a):=-{\lim\inf}_{n\to\infty}\log_{q_n}|a-(p_n/q_n)|\) wird als Irrationalitätsmaß (oder -exponent) von \(a\) bezeichnet. Schließlich wird \(C(a):=\inf_{(c_n)\in\mathbb{N}^\mathbb{N}} I([a_1c_1,a_2,c_2,\dots])\) nach \textit{P. Erdős} [J. Math. Sci. 10, 1--7 (1975; Zbl 0372.10023)] betrachtet. Eines der Hauptergebnisse lautet \(C(a)= 2^{\lim\sup_{n\to\infty}\log_2\log_2a_n}+1\), wobei \(\log_20:=0\) gesetzt ist. Ein zweites läßt sich so beschreiben: Hat man \(\lim_{n\to\infty}a_n^{1/K^n}\in]1,\infty[\) für ein reelles \(K>1\), so gilt \(I(a)=K+1\) und \(a\) ist transzendent. Kombination beider Resultate führt zu Folgendem: Seien \(K,a_1\in\mathbb{N}, K\geq2, a_{n+1}:=a_n^K+n!\) für jedes \(n\in\mathbb{N}\) und \(a:=[a_1,a_2,\dots]\), so ist \(I(a)=C(a)=K+1\) und \(a\) eine transzendente Zahl.
reviewed by: Peter Bundschuh
Mathematics Subject Classification ID: 11J82
Mathematics Subject Classification ID: 11A55
zbMATH DE Number: 5717801
MaRDI profile type: Publication
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full work available at URL: 0023608x-2009-002
OpenAlex ID: W2029601849
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description: scientific article; zbMATH DE number 5717801