Summary: Algebraic irrational binary numbers cannot be fixed points of
non­trivial constant length or primitive morphisms
Jean­Paul Allouche
CNRS, LRI
B“atiment 490
F­91405 Orsay Cedex (France)
allouche@lri.fr
Luca Q. Zamboni
Department of Mathematics
University of North Texas
Denton, Texas, TX 76203­5116 (USA)
lqz0001@jove.acs.unt.edu
May 2, 1997
Abstract
We prove that a positive real number whose binary expansion is a fixed point of a morphism
on the alphabet f0; 1g that is either of constant length – 2 or primitive, is either rational or
transcendental.
1 Introduction
How random are the digits of an algebraic irrational number in a given base? A common conjectured
answer to this vague question is that these digits are ``really random''. For example, define a normal

Source: Allouche, Jean-Paul - Laboratoire de Recherche en Informatique, Université de Paris-Sud 11

Collections: Computer Technologies and Information Sciences