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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...

An elementary proof that almost all real numbers are normal.

Filip, Ferdinánd, Šustek, Jan (2010)

Acta Universitatis Sapientiae. Mathematica

An iterated logarithm type theorem for the largest coefficient in continued fractions

János Galambos (1974)

Acta Arithmetica

Application de la théorie des algèbres de mesures à l'étude des mesures spectrales.

M. QUEFFELEC (1978/1979)

Seminaire de Théorie des Nombres de Bordeaux

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 $x^{2} - m x - n$ , $m \geq n \geq 1$ , 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 $β = τ = \frac{1}{2} (1 + \sqrt{5})$ , the golden ratio. For such $β$ , we determine the exact bound on the number of fractional digits...

Basic properties of shift radix systems.

Akiyama, Shigeki, Borbély, Tibor, Brunotte, Horst, Pethő, Attila, Thuswaldner, Jörg M. (2006)

Acta Mathematica Academiae Paedagogicae Nyí regyháziensis. New Series [electronic only]

Bemerkung über die Verteilung von Ziffern in Cantorschen Reihen

Tibor Šalát (1973)

Czechoslovak Mathematical Journal

Beta expansion of Salem numbers approaching Pisot numbers with the finiteness property

Hachem Hichri (2015)

Acta Arithmetica

It is already known that all Pisot numbers are beta numbers, but for Salem numbers this was proved just for the degree 4 case. In 1945, R. Salem showed that for any Pisot number θ we can construct a sequence of Salem numbers which converge to θ. In this short note, we give some results on the beta expansion for infinitely many sequences of Salem numbers obtained by this construction.

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