On the Normality of Arithmetical Constants

Bailey and Crandall [4] recently formulated “Hypothesis A”, a general principle to explain the (conjectured) normality of the binary expansion of constants like π and other related numbers, or more generally the base b expansion of such constants for an integer b ≥ 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. AMS Subject Classification(2000): 11K16 (Primary) 11A63, 28D05, 37E05 (Secondary)