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Real numbers with bounded partial quotients: A survey. (English) Zbl 0753.11006

In a regular continued fraction expansion \(x=[a_ 0,a_ 1,a_ 2,\ldots]\) of real numbers \(x\), where \(a_ 0\) is integer and \(a_ j\), \(j\geq 1\), are positive integers, the partial quotients \(a_ j\) are unbounded for almost all \(x\) with respect to Lebesgue measure. In fact, an accurate formula is available on the growth of the maximum of \(a_ 1,a_ 2,\ldots,a_ n\) valid for almost all \(x\) [see J. Galambos, Acta Arith. 25, 359-364 (1974; Zbl 0255.60033)]. Yet, continued fractions with bounded \(a_ j\) play an important role in a variety of fields. The author surveys such applications.

MSC:

11A55 Continued fractions
11K50 Metric theory of continued fractions
11-02 Research exposition (monographs, survey articles) pertaining to number theory
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