227 research outputs found
Minimal weight expansions in Pisot bases
For applications to cryptography, it is important to represent numbers with a
small number of non-zero digits (Hamming weight) or with small absolute sum of
digits. The problem of finding representations with minimal weight has been
solved for integer bases, e.g. by the non-adjacent form in base~2. In this
paper, we consider numeration systems with respect to real bases which
are Pisot numbers and prove that the expansions with minimal absolute sum of
digits are recognizable by finite automata. When is the Golden Ratio,
the Tribonacci number or the smallest Pisot number, we determine expansions
with minimal number of digits and give explicitely the finite automata
recognizing all these expansions. The average weight is lower than for the
non-adjacent form
Parallel addition in non-standard numeration systems
We consider numeration systems where digits are integers and the base is an
algebraic number such that and satisfies a
polynomial where one coefficient is dominant in a certain sense. For this class
of bases , we can find an alphabet of signed-digits on which addition is
realizable by a parallel algorithm in constant time. This algorithm is a kind
of generalization of the one of Avizienis. We also discuss the question of
cardinality of the used alphabet, and we are able to modify our algorithm in
order to work with a smaller alphabet. We then prove that satisfies
this dominance condition if and only if it has no conjugate of modulus 1. When
the base is the Golden Mean, we further refine the construction to
obtain a parallel algorithm on the alphabet . This alphabet cannot
be reduced any more
Rational numbers with purely periodic -expansion
We study real numbers with the curious property that the
-expansion of all sufficiently small positive rational numbers is purely
periodic. It is known that such real numbers have to be Pisot numbers which are
units of the number field they generate. We complete known results due to
Akiyama to characterize algebraic numbers of degree 3 that enjoy this property.
This extends results previously obtained in the case of degree 2 by Schmidt,
Hama and Imahashi. Let denote the supremum of the real numbers
in such that all positive rational numbers less than have a
purely periodic -expansion. We prove that is irrational
for a class of cubic Pisot units that contains the smallest Pisot number
. This result is motivated by the observation of Akiyama and Scheicher
that is surprisingly close to 2/3
Palindromic complexity of infinite words associated with simple Parry numbers
A simple Parry number is a real number \beta>1 such that the R\'enyi
expansion of 1 is finite, of the form d_\beta(1)=t_1...t_m. We study the
palindromic structure of infinite aperiodic words u_\beta that are the fixed
point of a substitution associated with a simple Parry number \beta. It is
shown that the word u_\beta contains infinitely many palindromes if and only if
t_1=t_2= ... =t_{m-1} \geq t_m. Numbers \beta satisfying this condition are the
so-called confluent Pisot numbers. If t_m=1 then u_\beta is an Arnoux-Rauzy
word. We show that if \beta is a confluent Pisot number then P(n+1)+ P(n) =
C(n+1) - C(n)+ 2, where P(n) is the number of palindromes and C(n) is the
number of factors of length n in u_\beta. We then give a complete description
of the set of palindromes, its structure and properties.Comment: 28 pages, to appear in Annales de l'Institut Fourie
On univoque Pisot numbers
We study Pisot numbers which are univoque, i.e., such that
there exists only one representation of 1 as , with . We prove in particular that there
exists a smallest univoque Pisot number, which has degree 14. Furthermore we
give the smallest limit point of the set of univoque Pisot numbers.Comment: Accepted by Mathematics of COmputatio
-block parallel addition versus -block parallel addition in non-standard numeration systems
Parallel addition in integer base is used for speeding up multiplication and
division algorithms. -block parallel addition has been introduced by
Kornerup in 1999: instead of manipulating single digits, one works with blocks
of fixed length . The aim of this paper is to investigate how such notion
influences the relationship between the base and the cardinality of the
alphabet allowing parallel addition. In this paper, we mainly focus on a
certain class of real bases --- the so-called Parry numbers. We give lower
bounds on the cardinality of alphabets of non-negative integer digits allowing
block parallel addition. By considering quadratic Pisot bases, we are able to
show that these bounds cannot be improved in general and we give explicit
parallel algorithms for addition in these cases. We also consider the
-bonacci base, which satisfies the equation . If in a base being a -bonacci number -block parallel
addition is possible on the alphabet , then ; on the other hand, there exists a such that -block
parallel addition in this base is possible on the alphabet , which
cannot be reduced. In particular, addition in the Tribonacci base is -block
parallel on alphabet .Comment: 21 page
FACTOR AND PALINDROMIC COMPLEXITY OF THUE-MORSE’S AVATARS
Two infinite words that are connected with some significant univoque numbers are studied. It is shown that their factor and palindromic complexities almost coincide with the factor and palindromic complexities of the famous Thue-Morse word
Negative bases and automata
We study expansions in non-integer negative base -{\beta} introduced by Ito
and Sadahiro. Using countable automata associated with (-{\beta})-expansions,
we characterize the case where the (-{\beta})-shift is a system of finite type.
We prove that, if {\beta} is a Pisot number, then the (-{\beta})-shift is a
sofic system. In that case, addition (and more generally normalization on any
alphabet) is realizable by a finite transducer. We then give an on-line
algorithm for the conversion from positive base {\beta} to negative base
-{\beta}. When {\beta} is a Pisot number, the conversion can be realized by a
finite on-line transducer
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