Metrical theory for $α$-Rosen fractions

Cor Kraaikamp; Karma Dajani; Wolfgang Steiner

doi:10.4171/jems/181

JournalsjemsVol. 11, No. 6pp. 1259–1283

Metrical theory for $α$ -Rosen fractions

Cor Kraaikamp
Delft University of Technology, Netherlands
Karma Dajani
Universiteit Utrecht, Netherlands
Wolfgang Steiner
Université Paris 7 Denis Diderot, France

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Abstract

The Rosen fractions form an inﬁnite family which generalizes the nearest-integer continued fractions. In this paper we introduce a new class of continued fractions related to the Rosen fractions, the α-Rosen fractions. The metrical properties of these α-Rosen fractions are studied.

We ﬁnd planar natural extensions for the associated interval maps, and show that their domains of deﬁnition are closely related to the domains of the ‘classical’ Rosen fractions. This uniﬁes and generalizes results of diophantine approximation from the literature.

Cite this article

Cor Kraaikamp, Karma Dajani, Wolfgang Steiner, Metrical theory for $α$ -Rosen fractions. J. Eur. Math. Soc. 11 (2009), no. 6, pp. 1259–1283

DOI 10.4171/JEMS/181