All a b c d e f g h i j k l m n o p q r s t u v w x y z Other

Page 1 Next

Displaying 1 – 20 of 155

Showing per page

Abstract β -expansions and ultimately periodic representations

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 ℚ ( β ) if the dominating eigenvalue β > 1 of the automaton accepting the language is a Pisot number. Moreover, if β is neither a Pisot nor a Salem number, then there exist points in ℚ ( β ) which do not have any ultimately periodic representation.

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...

Asymptotic nature of higher Mahler measure

(2014)

Acta Arithmetica

We consider Akatsuka’s zeta Mahler measure as a generating function of the higher Mahler measure m k ( P ) of a polynomial P , where m k ( P ) is the integral of l o g k | P | over the complex unit circle. Restricting ourselves to P(x) = x - r with |r| = 1 we show some new asymptotic results regarding m k ( P ) , in particular | m k ( P ) | / k ! → 1 / π as k → ∞.

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.

Currently displaying 1 – 20 of 155

Page 1 Next