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2001 On the Normality of Arithmetical Constants
Jeffrey C. Lagarias
Experiment. Math. 10(3): 355-368 (2001).

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.

Citation

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Jeffrey C. Lagarias. "On the Normality of Arithmetical Constants." Experiment. Math. 10 (3) 355 - 368, 2001.

Information

Published: 2001
First available in Project Euclid: 25 November 2003

zbMATH: 1015.11036
MathSciNet: MR1917424

Subjects:
Primary: 11K16
Secondary: 11A63 , 28D05 , 37E05

Keywords: $G$-functions , dynamical systems , Invariant measures , polylogarithms , radix expansions

Rights: Copyright © 2001 A K Peters, Ltd.

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Vol.10 • No. 3 • 2001
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