1,846 research outputs found
Number representation using generalized -transformation
We study non-standard number systems with negative base . Instead of
the Ito-Sadahiro definition, based on the transformation of the
interval into itself, we
suggest a generalization using an interval with . Such
generalization may eliminate certain disadvantages of the Ito-Sadahiro system.
We focus on the description of admissible digit strings and their periodicity.Comment: 22 page
A Probabilistic Approach to Generalized Zeckendorf Decompositions
Generalized Zeckendorf decompositions are expansions of integers as sums of
elements of solutions to recurrence relations. The simplest cases are base-
expansions, and the standard Zeckendorf decomposition uses the Fibonacci
sequence. The expansions are finite sequences of nonnegative integer
coefficients (satisfying certain technical conditions to guarantee uniqueness
of the decomposition) and which can be viewed as analogs of sequences of
variable-length words made from some fixed alphabet. In this paper we present a
new approach and construction for uniform measures on expansions, identifying
them as the distribution of a Markov chain conditioned not to hit a set. This
gives a unified approach that allows us to easily recover results on the
expansions from analogous results for Markov chains, and in this paper we focus
on laws of large numbers, central limit theorems for sums of digits, and
statements on gaps (zeros) in expansions. We expect the approach to prove
useful in other similar contexts.Comment: Version 1.0, 25 pages. Keywords: Zeckendorf decompositions, positive
linear recurrence relations, distribution of gaps, longest gap, Markov
processe
Substitutions over infinite alphabet generating (-\beta)-integers
This contribution is devoted to the study of positional numeration systems
with negative base introduced by Ito and Sadahiro in 2009, called
(-\beta)-expansions. We give an admissibility criterion for more general case
of (-\beta)-expansions and discuss the properties of the set of
(-\beta)-integers. We give a description of distances within this set and show
that this set can be coded by an infinite word over an infinite alphabet, which
is a fixed point of a non-erasing non-trivial morphism.Comment: In Proceedings WORDS 2011, arXiv:1108.341
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)
Numeration Systems: a Link between Number Theory and Formal Language Theory
We survey facts mostly emerging from the seminal results of Alan Cobham
obtained in the late sixties and early seventies. We do not attempt to be
exhaustive but try instead to give some personal interpretations and some
research directions. We discuss the notion of numeration systems, recognizable
sets of integers and automatic sequences. We briefly sketch some results about
transcendence related to the representation of real numbers. We conclude with
some applications to combinatorial game theory and verification of
infinite-state systems and present a list of open problems.Comment: 21 pages, 3 figures, invited talk DLT'201
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
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