A class of transcendental numbers with bounded partial quotients. (English) Zbl 0693.10028

Number theory and applications, Proc. NATO ASI, Banff/Can. 1988, NATO ASI Ser., Ser. C 265, 356-371 (1989).
[For the entire collection see Zbl 0676.00005.]
For real \(x\geq 0\), let S(x) be the continued fraction \([b_ 0,b_ 1,b_ 2,...]\) with \(b_ n=1+([nx] mod 2)\). Thus S(x) is in the class F(2) of continued fractions with partial quotients bounded by 2. Theorem: Suppose that x is irrational and that infinitely many convergents to x with even index have even numerators. Then S(x) is transcendental. For the proof it is shown that sufficiently many convergents to S(x) are close enough to apply W. M. Schmidt’s generalization [Acta. Math. 119, 27-50 (1967; Zbl 0173.048)] of the Thue-Siegel-Roth theorem.
Reviewer: G.Köhler


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