A Class of Discrete Spectra of Non-Pisot Numbers
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Dragan Stankov (2008)
Publications de l'Institut Mathématique
Barat, Guy, Frougny, Christiane, Pethő, Attila (2005)
Integers
Shigeki Akiyama, Taizo Sadahiro (1998)
Acta Mathematica et Informatica Universitatis Ostraviensis
Michel Rigo, Wolfgang Steiner (2005)
Journal de Théorie des Nombres de Bordeaux
For abstract numeration systems built on exponential regular languages (including those coming from substitutions), we show that the set of real numbers having an ultimately periodic representation is if the dominating eigenvalue of the automaton accepting the language is a Pisot number. Moreover, if is neither a Pisot nor a Salem number, then there exist points in which do not have any ultimately periodic representation.
Yves Meyer (1971/1972)
Séminaire Delange-Pisot-Poitou. Théorie des nombres
C. Lloyd-Smith (1985)
Acta Arithmetica
Xavier Stefani (1968/1969)
Séminaire Delange-Pisot-Poitou. Théorie des nombres
Mostapha Bouhamza (1984)
Acta Arithmetica
Nikita Sidorov (2002)
Acta Arithmetica
Paul Voutier (1996)
Acta Arithmetica
A. Bazylewicz (1988)
Acta Arithmetica
Anne Bertrand-Mathis (1985)
Annales de l'institut Fourier
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.
B. Sury (1992)
Manuscripta mathematica
Zuzana Masáková, Tomáš Vávra (2011)
Kybernetika
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...
L. S. Guimond, Z. Masáková, E. Pelantová (2004)
Acta Arithmetica
(2014)
Acta Arithmetica
We consider Akatsuka’s zeta Mahler measure as a generating function of the higher Mahler measure of a polynomial where is the integral of over the complex unit circle. Restricting ourselves to P(x) = x - r with |r| = 1 we show some new asymptotic results regarding , in particular as k → ∞.
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.
DoYong Kwon (2007)
Acta Arithmetica
Robert Kaufman (1974)
Studia Mathematica
Françoise Bertrandias (1963/1964)
Séminaire Dubreil. Algèbre et théorie des nombres
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