Metrical theory for α-Rosen fractions

Metrical Theory for Α-rosen Fractions

Abstract. The Rosen fractions form an infinite family which generalizes the nearestinteger 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 these regions are clos...

Metrical Theory for Farey Continued Fractions

By making fundamental use of the Farey shift map and employing infinite (but σ-finite) measures together with the Chacon-Ornstein ergodic theorem it is possible to find new metrical results for continued fractions. Moreover this offers a unified approach to several existing theorems. The application of ergodic theory to the study of continued fractions began with the Gauss transformation, G: [0...

Metrical Theory for Optimal Continued Fractions

where Ed= fl, bkEZ2,, k 2 1, and with some constraints on bk and sk. Usually we will assume that x is irrational, and thus that the expansion (1.1) is infinite. A special case of an SRCF is the regular continuedfraction, RCF, which is obtained by taking ck = 1 for every k in (1.1). The aim in introducing the OCF was to optimize two things simultaneously. In the first place one wishes the conver...

Transcendence with Rosen Continued Fractions

We give the first transcendence results for the Rosen continued fractions. Introduced over half a century ago, these fractions expand real numbers in terms of certain algebraic numbers.

Rosen Representation Theory Notes