Metrical theory for $\alpha$-Rosen fractions

Karma Dajani (1) , Cor Kraaikamp (2) , Wolfgang Steiner (3)

1 Department of Mathematics
2 Electrical Engineering, Mathematics and Computer Science
3 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications

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 $\alpha$-Rosen fractions. The metrical properties of these $\alpha$-Rosen fractions are studied. We find planar natural extensions for the associated interval maps, and show that these regions are closely related to similar region for the 'classical' Rosen fraction. This allows us to unify and generalize results of diophantine approximation from the literature.

Mots clés: Rosen fractions; natural extension; Diophantine approximation

Domaines: Théorie des nombres [math.NT]; Systèmes dynamiques [math.DS]

Karma Dajani, Cor Kraaikamp, Wolfgang Steiner. Metrical theory for $\alpha$-Rosen fractions. Journal of the European Mathematical Society, 2009, 11 (6), pp.1259-1283. ⟨hal-00131276v2⟩