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Arithmetics in numeration systems with negative quadratic base

Zuzana MasákováTomáš 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...

Morphisms fixing words associated with exchange of three intervals

Petr AmbrožZuzana MasákováEdita Pelantová — 2010

RAIRO - Theoretical Informatics and Applications

We consider words coding exchange of three intervals with permutation (3,2,1), here called 3iet words. Recently, a characterization of substitution invariant 3iet words was provided. We study the opposite question: what are the morphisms fixing a 3iet word? We reveal a narrow connection of such morphisms and morphisms fixing Sturmian words using the new notion of amicability.

Complexity of infinite words associated with beta-expansions

Christiane FrougnyZuzana MasákováEdita Pelantová — 2004

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

We study the complexity of the infinite word u β associated with the Rényi expansion of 1 in an irrational base β > 1 . When β is the golden ratio, this is the well known Fibonacci word, which is sturmian, and of complexity ( n ) = n + 1 . For β such that d β ( 1 ) = t 1 t 2 t m is finite we provide a simple description of the structure of special factors of the word u β . When t m = 1 we show that ( n ) = ( m - 1 ) n + 1 . In the cases when t 1 = t 2 = = t m - 1 or t 1 > max { t 2 , , t m - 1 } we show that the first difference of the complexity function ( n + 1 ) - ( n ) takes value in { m - 1 , m } for every n , and consequently we determine...

Complexity of infinite words associated with beta-expansions

Christiane FrougnyZuzana MasákováEdita Pelantová — 2010

RAIRO - Theoretical Informatics and Applications

We study the complexity of the infinite word associated with the Rényi expansion of in an irrational base . When is the golden ratio, this is the well known Fibonacci word, which is Sturmian, and of complexity . For such that is finite we provide a simple description of the structure of special factors of the word . When =1 we show that . In the cases when or max} we show that the first difference of the complexity function takes value in for every , and consequently we determine the complexity...

Greedy and lazy representations in negative base systems

Tomáš HejdaZuzana MasákováEdita Pelantová — 2013

Kybernetika

We consider positional numeration systems with negative real base - β , where β > 1 , and study the extremal representations in these systems, called here the greedy and lazy representations. We give algorithms for determination of minimal and maximal ( - β ) -representation with respect to the alternate order. We also show that both extremal representations can be obtained as representations in the positive base β 2 with a non-integer alphabet. This enables us to characterize digit sequences admissible as greedy...

Combinatorial properties of infinite words associated with cut-and-project sequences

Louis-Sébastien GuimondZuzana MasákováEdita Pelantová — 2003

Journal de théorie des nombres de Bordeaux

The aim of this article is to study certain combinatorial properties of infinite binary and ternary words associated to cut-and-project sequences. We consider here the cut-and-project scheme in two dimensions with general orientation of the projecting subspaces. We prove that a cut-and-project sequence arising in such a setting has always either two or three types of distances between adjacent points. A cut-and-project sequence thus determines in a natural way a symbolic sequence (infinite word)...

Palindromic complexity of infinite words associated with simple Parry numbers

Petr AmbrožZuzana MasákováEdita PelantováChristiane Frougny — 2006

Annales de l’institut Fourier

A simple Parry number is a real number β > 1 such that the Rényi expansion of 1 is finite, of the form d β ( 1 ) = t 1 t m . We study the palindromic structure of infinite aperiodic words u β that are the fixed point of a substitution associated with a simple Parry number β . It is shown that the word u β contains infinitely many palindromes if and only if t 1 = t 2 = = t m - 1 t m . Numbers β satisfying this condition are the so-called Pisot numbers. If t m = 1 then u β is an Arnoux-Rauzy word. We show that if β is a confluent Pisot number then 𝒫 ( n + 1 ) + 𝒫 ( n ) = 𝒞 ( n + 1 ) - 𝒞 ( n ) + 2 , where...

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