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.
@article{1069786344,
author = {Lagarias, Jeffrey C.},
title = {On the Normality of Arithmetical Constants},
journal = {Experiment. Math.},
volume = {10},
number = {3},
year = {2001},
pages = { 355-368},
language = {en},
url = {http://dml.mathdoc.fr/item/1069786344}
}
Lagarias, Jeffrey C. On the Normality of Arithmetical Constants. Experiment. Math., Tome 10 (2001) no. 3, pp. 355-368. http://gdmltest.u-ga.fr/item/1069786344/