Search

We present two (inequivalent) polynomial continued fraction representations of the number e π with all their elements in Q ; no such representation was seemingly known before. More generally, a similar result for e r π is obtained for every r ∈ R such that r 2 ∈ Q . The proof uses a classical polynomial continued fraction representation of α β , for | arg ⁡ (α) | < π and β ∈ C ∖ Z , of which we present a new proof that enables us to obtain the exact rate of convergence of the convergents of the continued fraction for e π . We also deduce some consequences of arithmetic interest concerning the elements of certain polynomial continued fraction representations of the (transcendental) Gel'fond-Schneider numbers α β , where α ∈ Q ¯ ∖ { 0 , 1 } and β ∈ Q ¯ ∖ Q , where Q ¯ is the field of algebraic numbers, embedded into C . [ABSTRACT FROM AUTHOR]

Let K( an/ bn) be a continued fraction with elements ( an, bn) picked randomly and independently from $(\mathbb{C}\setminus\{0\})\times\mathbb{C}$ according to some probability distribution μ. We find sufficient conditions on μ for K( an/ bn) to converge with probability 1 or to be restrained with probability 1. More generally, we also consider μ-random sequences { τn} of independent Möbius transformations and find sufficient conditions for $\{\tau_{1}\circ\tau_{2}\circ\cdots\circ\tau_{n}\}_{n=1}^{\infty}$ to converge or be restrained with probability 1. The analysis is based on an important paper by Furstenberg. [ABSTRACT FROM AUTHOR]