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An alternative construction of normal numbers

Edgardo Ugalde (2000)

Journal de théorie des nombres de Bordeaux

A new class of b -adic normal numbers is built recursively by using Eulerian paths in a sequence of de Bruijn digraphs. In this recursion, a path is constructed as an extension of the previous one, in such way that the b -adic block determined by the path contains the maximal number of different b -adic subblocks of consecutive lengths in the most compact arrangement. Any source of redundancy is avoided at every step. Our recursive construction is an alternative to the several well-known concatenative...

Applications de la notion d'entropie au développement d'un nombre réel dans une base de Pisot

Anne Bertrand-Mathis (1985)

Annales de l'institut Fourier

Soit θ un nombre de Pisot de degré s  ; nous avons montré précédemment que l’endomorphisme du tore T s dont θ est valeur propre est facteur du θ -shift bilatéral par une application continue q s  ; nous prouvons ici (théorème 1) que l’application q s conserve l’entropie de toute mesure invariante sur le θ -shift. Ceci permet de définir l’entropie d’un nombre dans la base θ et d’en étudier la stabilité. Nous généralisons également des résultats de Kamae, Rauzy et Bernay.

Arithmetics in numeration systems with negative quadratic base

Zuzana Masáková, Tomáš Vávra (2011)

Kybernetika

We consider positional numeration system with negative base - β , as introduced by Ito and Sadahiro. In particular, we focus on arithmetical properties of such systems when β is a quadratic Pisot number. We study a class of roots β > 1 of polynomials x 2 - m x - n , m n 1 , and show that in this case the set Fin ( - β ) of finite ( - β ) -expansions is closed under addition, although it is not closed under subtraction. A particular example is β = τ = 1 2 ( 1 + 5 ) , the golden ratio. For such β , we determine the exact bound on the number of fractional digits...

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