A combinatorial method for construction normal numbers.
A conjecture of Erdös on continued fractions
A generalization of the Davenport-Erdős construction of normal numbers
We extend the Davenport and Erdős construction of normal numbers to the case.
A general arithmetic construction of transcendental non-Liouville normal numbers from rational fractions
A generalization of Hasse's generalization of the Syracuse algorithm
A Markov process underlying the generalized Syracuse algorithm
A propos du théorème de Cassels-Schmidt concernant de nombres normaux
A self-similar tiling generated by the minimal Pisot number
Abschätzung der Hausdorffdimension für Mengen mit vorgeschriebenen Häufigkeiten der Ziffern.
Algebraic independence of real numbers with low density of nonzero digits
All Generalized Morse-Sequences are Loosely Bernoulli.
An alternative construction of normal numbers
A new class of -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 -adic block determined by the path contains the maximal number of different -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.
An iterated logarithm type theorem for the largest coefficient in continued fractions
Application de la théorie des algèbres de mesures à l'étude des mesures spectrales.
Applications de la notion d'entropie au développement d'un nombre réel dans une base de Pisot
Soit un nombre de Pisot de degré ; nous avons montré précédemment que l’endomorphisme du tore dont est valeur propre est facteur du -shift bilatéral par une application continue ; nous prouvons ici (théorème 1) que l’application conserve l’entropie de toute mesure invariante sur le -shift. Ceci permet de définir l’entropie d’un nombre dans la base et d’en étudier la stabilité. Nous généralisons également des résultats de Kamae, Rauzy et Bernay.
Arithmetics in numeration systems with negative quadratic base
We consider positional numeration system with negative base , as introduced by Ito and Sadahiro. In particular, we focus on arithmetical properties of such systems when is a quadratic Pisot number. We study a class of roots of polynomials , , and show that in this case the set of finite -expansions is closed under addition, although it is not closed under subtraction. A particular example is , the golden ratio. For such , we determine the exact bound on the number of fractional digits...
Basic properties of shift radix systems.
Bemerkung über die Verteilung von Ziffern in Cantorschen Reihen
Beta expansion of Salem numbers approaching Pisot numbers with the finiteness property
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.