Continued Fractions with Increasing Digits

A Dimension Gap for Continued Fractions with Independent Digits

On Transfer Operators for Continued Fractions with Restricted Digits

For I ⊂ N, let ΛI denote those numbers in the unit interval whose continued fraction digits all lie in I. Define the corresponding transfer operator LI,βf(z) = ∑ n∈I ( 1 n+z )2β f ( 1 n+z ) for Re(β) > max(0, θI), where Re(β) = θI is the abscissa of convergence of the series ∑ n∈I n −2β . When acting on a certain Hilbert space HI,β , we show that the operator LI,β is conjugate to an integral op...

Continued Fractions with Multiple Limits

For integers m ≥ 2, we study divergent continued fractions whose numerators and denominators in each of the m arithmetic progressions modulo m converge. Special cases give, among other things, an infinite sequence of divergence theorems, the first of which is the classical Stern-Stolz theorem. We give a theorem on a class of Poincaré type recurrences which shows that they tend to limits when th...

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.

Generalized Continued Logarithms and Related Continued Fractions

We study continued logarithms as introduced by Bill Gosper and studied by J. Borwein et. al.. After providing an overview of the type I and type II generalizations of binary continued logarithms introduced by Borwein et. al., we focus on a new generalization to an arbitrary integer base b. We show that all of our so-called type III continued logarithms converge and all rational numbers have fin...