Metrical theory for $α$-Rosen fractions

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.