Continued fractional measure of irrationality. (English) Zbl 1221.11157

Hančl, Jaroslav; Matala-Aho, Tapani; Pulcerová, Simona

Kyoto J. Math. 50, No. 1, 33-40 (2010).

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 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.

Reviewer: Peter Bundschuh (Köln)

MSC:

11J82	Measures of irrationality and of transcendence
11A55	Continued fractions

Citations:

Zbl 0372.10023

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