On the Normality of Arithmetical Constants

Abstract

Bailey and Crandall recently formulated "Hypothesis A'', a general principle to explain the (conjectured) normality of the binary expansion of constants like $\pi$ and other related numbers, or more generally the base $b$ expansion of such constants for an integer $b \geq 2$. This paper shows that a basic mechanism underlying their principle, which is a relation between single orbits of two discrete dynamical systems, holds for a very general class of representations of numbers. This general class includes numbers for which the conclusion of Hypothesis A is not true. The paper also relates the particular class of arithmetical constants treated by Bailey and Crandall to special values of $G$-functions, and points out an analogy of Hypothesis A with Furstenberg's conjecture on invariant measures.