22 research outputs found
Addition and multiplication of beta-expansions in generalized Tribonacci base
We study properties of β-numeration systems, where β > 1 is the real root of the polynomial x3 - mx2 - x - 1, m ∈ ℕ, m ≥ 1. We consider arithmetic operations on the set of β-integers, i.e., on the set of numbers whose greedy expansion in base β has no fractional part. We show that the number of fractional digits arising under addition of β-integers is at most 5 for m ≥ 3 and 6 for m = 2, whereas under multiplication it is at most 6 for all m ≥ 2. We thus generalize the results known for Tribonacci numeration system, i.e., for m = 1. We summarize the combinatorial properties of infinite words naturally defined by β-integers. We point out the differences between the structure of β-integers in cases m = 1 and m ≥ 2
Arithmetics in β-numeration
Analysis of AlgorithmsInternational audienceThe β-numeration, born with the works of Rényi and Parry, provides a generalization of the notions of integers, decimal numbers and rational numbers by expanding real numbers in base β, where β>1 is not an integer. One of the main differences with the case of numeration in integral base is that the sets which play the role of integers, decimal numbers and rational numbers in base β are not stable under addition or multiplication. In particular, a fractional part may appear when one adds or multiplies two integers in base β. When β is a Pisot number, which corresponds to the most studied case, the lengths of the finite fractional parts that may appear when one adds or multiplies two integers in base β are bounded by constants which only depend on β. We prove that, for any Perron number β, the set of finite or ultimately periodic fractional parts of the sum, or the product, of two integers in base β is finite. Additionally, we prove that it is possible to compute this set for the case of addition when β is a Parry number. As a consequence, we deduce that, when β is a Perron number, there exist bounds, which only depend on β, for the lengths of the finite fractional parts that may appear when one adds or multiplies two integers in base β. Moreover, when β is a Parry number, the bound associated with the case of addition can be explicitly computed
Non-Standard Numeration Systems
We study some properties of non-standard numeration systems with an irrational base ß >1, based on the so-called beta-expansions of real numbers [1]. We discuss two important properties of these systems, namely the Finiteness property, stating whether the set of finite expansions in a given system forms a ring, and then the problem of fractional digits arising under arithmetic operations with integers in a given system. Then we introduce another way of irrational representation of numbers, slightly different from classical beta-expansions. Here we restrict ourselves to one irrational base – the golden mean ? – and we study the Finiteness property again.
-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
Finite beta-expansions with negative bases
The finiteness property is an important arithmetical property of
beta-expansions. We exhibit classes of Pisot numbers having the
negative finiteness property, that is the set of finite -expansions
is equal to . For a class of numbers including the
Tribonacci number, we compute the maximal length of the fractional parts
arising in the addition and subtraction of -integers. We also give
conditions excluding the negative finiteness property
Dynamical Directions in Numeration
International audienceWe survey definitions and properties of numeration from a dynamical point of view. That is we focuse on numeration systems, their associated compactifications, and the dynamical systems that can be naturally defined on them. The exposition is unified by the notion of fibred numeration system. A lot of examples are discussed. Various numerations on natural, integral, real or complex numbers are presented with a special attention payed to beta-numeration and its generalisations, abstract numeration systems and shift radix systems. A section of applications ends the paper
Fractal tiles associated with shift radix systems
Shift radix systems form a collection of dynamical systems depending on a
parameter which varies in the -dimensional real vector space.
They generalize well-known numeration systems such as beta-expansions,
expansions with respect to rational bases, and canonical number systems.
Beta-numeration and canonical number systems are known to be intimately related
to fractal shapes, such as the classical Rauzy fractal and the twin dragon.
These fractals turned out to be important for studying properties of expansions
in several settings. In the present paper we associate a collection of fractal
tiles with shift radix systems. We show that for certain classes of parameters
these tiles coincide with affine copies of the well-known tiles
associated with beta-expansions and canonical number systems. On the other
hand, these tiles provide natural families of tiles for beta-expansions with
(non-unit) Pisot numbers as well as canonical number systems with (non-monic)
expanding polynomials. We also prove basic properties for tiles associated with
shift radix systems. Indeed, we prove that under some algebraic conditions on
the parameter of the shift radix system, these tiles provide
multiple tilings and even tilings of the -dimensional real vector space.
These tilings turn out to have a more complicated structure than the tilings
arising from the known number systems mentioned above. Such a tiling may
consist of tiles having infinitely many different shapes. Moreover, the tiles
need not be self-affine (or graph directed self-affine)
Combinatorial and Arithmetical Properties of Infinite Words Associated with Non-simple Quadratic Parry Numbers
We study arithmetical and combinatorial properties of -integers for
being the root of the equation . We determine with the accuracy of the maximal number of
-fractional positions, which may arise as a result of addition of two
-integers. For the infinite word coding distances between
consecutive -integers, we determine precisely also the balance. The word
is the fixed point of the morphism and . In the case the corresponding infinite word is
sturmian and therefore 1-balanced. On the simplest non-sturmian example with
, we illustrate how closely the balance and arithmetical properties of
-integers are related.Comment: 15 page
Shift Radix Systems - A Survey
Let be an integer and . The {\em shift radix system} is defined by has the {\em finiteness
property} if each is eventually mapped to
under iterations of . In the present survey we summarize
results on these nearly linear mappings. We discuss how these mappings are
related to well-known numeration systems, to rotations with round-offs, and to
a conjecture on periodic expansions w.r.t.\ Salem numbers. Moreover, we review
the behavior of the orbits of points under iterations of with
special emphasis on ultimately periodic orbits and on the finiteness property.
We also describe a geometric theory related to shift radix systems.Comment: 45 pages, 16 figure