Journal of the European Mathematical Society

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Volume 11, Issue 6, 2009, pp. 1259–1283
DOI: 10.4171/JEMS/181

Metrical theory for α-Rosen fractions

Karma Dajani[1], Cor Kraaikamp[2] and Wolfgang Steiner[3]

(1) Fac. Wiskunde en Informatica and MRI, Universiteit Utrecht, Budapestlaan 6, P.O. Box 88000, 3508 TA, Utrecht, Netherlands
(2) Control, Risk, Optimization, Systems and Stochasti, Delft University of Technology, Mekelweg 4, 2628 CD, Delft, Netherlands
(3) LIAFA. CNRS UMR 7089, Université Paris 7, Denis Diderot, Case 7014, 2 place Jussieu, 75251, Paris CEDEX 05, France

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.

Keywords: Rosen fractions, natural extension, Diophantine approximation

Dajani Karma, Kraaikamp Cor, Steiner Wolfgang: Metrical theory for α-Rosen fractions. J. Eur. Math. Soc. 11 (2009), 1259-1283. doi: 10.4171/JEMS/181