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Transcendence criteria for pairs of continued fractions. (English) Zbl 1123.11023

Several new transcendental criteria were made by the authors in [Acta Math. 195, No. 1, 1–20 (2005; Zbl 1195.11093), J. Reine Angew. Math. 606, 105–121 (2007; Zbl 1145.11054) and Ann. Inst. Fourier 57, No. 5, 1557–1574 (2007; Zbl 1126.11036)].
he purpose of this paper is to establish two extensions of some transcendence criteria for real numbers given by their continued fraction expansions. The authors adopt a slightly different point of view: rather than giving sufficient conditions ensuring the transcendence of a given number \(\alpha\), they take a pair \((\alpha, \alpha')\) of real numbers, and they prove that, under some condition, at least one of them is transcendental. The proofs rest on the Schmidt subspace theorem.

MSC:

11J81 Transcendence (general theory)
11J70 Continued fractions and generalizations
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