90 research outputs found
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
Purely periodic beta-expansions in the Pisot non-unit case
It is well known that real numbers with a purely periodic decimal expansion
are the rationals having, when reduced, a denominator coprime with 10. The aim
of this paper is to extend this result to beta-expansions with a Pisot base
beta which is not necessarily a unit: we characterize real numbers having a
purely periodic expansion in such a base; this characterization is given in
terms of an explicit set, called generalized Rauzy fractal, which is shown to
be a graph-directed self-affine compact subset of non-zero measure which
belongs to the direct product of Euclidean and p-adic spaces
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
Beta-expansions, natural extensions and multiple tilings associated with Pisot units
From the works of Rauzy and Thurston, we know how to construct (multiple)
tilings of some Euclidean space using the conjugates of a Pisot unit
and the greedy -transformation. In this paper, we consider different
transformations generating expansions in base , including cases where
the associated subshift is not sofic. Under certain mild conditions, we show
that they give multiple tilings. We also give a necessary and sufficient
condition for the tiling property, generalizing the weak finiteness property
(W) for greedy -expansions. Remarkably, the symmetric
-transformation does not satisfy this condition when is the
smallest Pisot number or the Tribonacci number. This means that the Pisot
conjecture on tilings cannot be extended to the symmetric
-transformation. Closely related to these (multiple) tilings are natural
extensions of the transformations, which have many nice properties: they are
invariant under the Lebesgue measure; under certain conditions, they provide
Markov partitions of the torus; they characterize the numbers with purely
periodic expansion, and they allow determining any digit in an expansion
without knowing the other digits
Linear recursive odometers and beta-expansions
The aim of this paper is to study the connection between different properties
related to -expansions. In particular, the relation between two
conditions, both ensuring pure discrete spectrum of the odometer, is analysed.
The first one is the so-called Hypothesis B for the -odometers and the
second one is denoted by (QM) and it has been introduced in the framework of
tilings associated to Pisot -numerations
Parametrization for a class of Rauzy Fractal
In this paper, we study a class of Rauzy fractals given by
the polynomial where is an integer. In particular we
give explicitly an automaton that generates the boundary of
and using an exotic numeration system we prove that is
homeomorphic to a topological disk
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