Continued fractions and transcendental numbers

Continued fractions and transcendental numbers

It is widely believed that the continued fraction expansion of every irrational algebraic number \(\alpha\) either is eventually periodic (and we know that this is the case if and only if \(\alpha\) is a quadratic irrational), or it contains arbitrarily large partial quotients. Apparently, this ques...

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Veröffentlicht in:arXiv.org 2005-11
Hauptverfasser: Adamczewski, Boris, Bugeaud, Yann, Davison, Les J L
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Combinatorial analysis
Quotients
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Davison, Les J L
description It is widely believed that the continued fraction expansion of every irrational algebraic number \(\alpha\) either is eventually periodic (and we know that this is the case if and only if \(\alpha\) is a quadratic irrational), or it contains arbitrarily large partial quotients. Apparently, this question was first considered by Khintchine. A preliminary step towards its resolution consists in providing explicit examples of transcendental continued fractions. The main purpose of the present work is to present new families of transcendental continued fractions with bounded partial quotients. Our results are derived thanks to new combinatorial transcendence criteria recently obtained by Adamczewski and Bugeaud.
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subjects Combinatorial analysis
Quotients
title Continued fractions and transcendental numbers
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