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Dynamical directions in numeration
 Ann. Inst. Fourier
"... Abstract. This survey aims at giving a consistent presentation of numeration from a dynamical viewpoint: we focus on numeration systems, their associated compactification, and dynamical systems that can be naturally defined on them. The exposition is unified by the fibred numeration system concept. ..."
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Abstract. This survey aims at giving a consistent presentation of numeration from a dynamical viewpoint: we focus on numeration systems, their associated compactification, and dynamical systems that can be naturally defined on them. The exposition is unified by the fibred numeration system concept. Many examples are discussed. Various numerations on rational integers, real or complex numbers are presented with special attention paid to βnumeration and its generalisations, abstract numeration systems and shift radix systems, as well as Gscales and odometers. A section of applications ends the paper. Le but de ce survol est d’aborder définitions et propriétés concernant la numération d’un point de vue dynamique: nous nous concentrons sur les systèmes de numération, leur compactification, et les systèmes dynamiques définis sur ces espaces. La notion de système de numération fibré unifie la présentation. De nombreux exemples sont étudiés. Plusieurs numérations sur les entiers naturels, relatifs, les nombres réels ou complexes sont présentées.
WORDS AND TRANSCENDENCE
, 2009
"... Abstract. Is it possible to distinguish algebraic from transcendental real numbers by considering the bary expansion in some base b � 2? In 1950, É. Borel suggested that the answer is no and that for any real irrational algebraic number x and for any base g � 2, the gary expansion of x should sati ..."
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Abstract. Is it possible to distinguish algebraic from transcendental real numbers by considering the bary expansion in some base b � 2? In 1950, É. Borel suggested that the answer is no and that for any real irrational algebraic number x and for any base g � 2, the gary expansion of x should satisfy some of the laws that are shared by almost all numbers. For instance, the frequency where a given finite sequence of digits occurs should depend only on the base and on the length of the sequence. We are very far from such a goal: there is no explicitly known example of a triple (g, a, x), where g � 3 is an integer, a a digit in {0,...,g − 1} and x a real irrational algebraic number, for which one can claim that the digit a occurs infinitely often in the gary expansion of x. Hence there is a huge gap between the established theory and the expected state of the art. However, some progress has been made recently, thanks mainly to clever use of Schmidt’s subspace theorem. We review some of these results. 1. Normal Numbers and Expansion of Fundamental Constants 1.1. Borel and Normal Numbers. In two papers, the first [28] published in 1909 and the second [29] in 1950, Borel studied the gary expansion of real numbers, where g � 2 is a positive integer. In his second paper, he suggested that this expansion for a real irrational algebraic number should satisfy some of the laws shared by almost all numbers, in the sense of Lebesgue measure. Let g � 2 be an integer. Any real number x has a unique expansion x = a−kg k + ···+ a−1g + a0 + a1g −1 + a2g −2 + ·· ·, where k � 0 is an integer and the ai for i � −k, namely the digits of x in the expansion in base g of x, belong to the set {0, 1,...,g − 1}. Uniqueness is subject to the condition that the sequence (ai)i�−k is not ultimately constant and equal to g − 1. We write this expansion x = a−k ···a−1a0.a1a2 ·· ·. in base 10 (decimal expansion), whereas