Experimental Mathematics

On the Normality of Arithmetical Constants

Jeffrey C. Lagarias

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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 information

Experiment. Math., Volume 10, Issue 3 (2001), 355-368.

First available in Project Euclid: 25 November 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11K16: Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. [See also 11A63]
Secondary: 11A63: Radix representation; digital problems {For metric results, see 11K16} 28D05: Measure-preserving transformations 37E05: Maps of the interval (piecewise continuous, continuous, smooth)

Dynamical systems invariant measures $G$-functions polylogarithms radix expansions


Lagarias, Jeffrey C. On the Normality of Arithmetical Constants. Experiment. Math. 10 (2001), no. 3, 355--368. https://projecteuclid.org/euclid.em/1069786344

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