arXiv subjects

E. Pelantová

Publications and source records attributed to E. Pelantová.

11 recordsLinked to original sources

arXiv2011

Numbers with integer expansion in the numeration system with negative base

In this paper, we study representations of real numbers in the positional numeration system with negative basis, as introduced by Ito and Sadahiro. We focus on the set $\Z_{-β}$ of numbers whose representation uses only non-negative powers of $-β$, the so-called $(-β)$-integers. We describe the distances between consecutive elements of $\Z_{-β}$. In case that this set is non-trivial we associate to $β$ an infinite word $\boldsymbol{v}_{-β}$ over an (in general infinite) alphabet. The self-similarity of $\Z_{-β}$, i.e., the property $-β\Z_{-β}\subset \Z_{-β}$, allows us to find a morphism under which $\boldsymbol{v}_{-β}$ is invariant. On the example of two cubic irrational bases $β$ we demonstrate the difference between Rauzy fractals generated by $(-β)$-integers and by $β$-integers.

math.NT↗

arXiv2010

Arithmetics in number systems with negative base

We study the numeration system with negative basis, introduced by Ito and Sadahiro. We focus on arithmetic operations in the set ${\rm Fin}(-β)$ and $\Z_{-β}$ of numbers having finite resp. integer $(-β)$-expansions. We show that ${\rm Fin}(-β)$ is trivial if $β$ is smaller than the golden ratio $\frac12(1+\sqrt5)$. For $β\geq\frac12(1+\sqrt5)$ we prove that ${\rm Fin}(-β)$ is a ring, only if $β$ is a Pisot or Salem number with no negative conjugates. We prove the conjecture of Ito and Sadahiro that ${\rm Fin}(-β)$ is a ring if $β$ is a quadratic Pisot number with positive conjugate. For quadratic Pisot units we determine the number of fractional digits that may appear when adding or multiplying two $(-β)$-integers.

math.NT↗

arXiv2010

Critical exponent of infinite words coding beta-integers associated with non-simple Parry numbers

In this paper, we study the critical exponent of infinite words $\ubeta$ coding $β$-integers for $β$ being a~non-simple Parry number. In other words, we investigate the maximal consecutive repetitions of factors that occur in the infinite word in question. We calculate also the ultimate critical exponent that expresses how long repetitions occur in the infinite word $\ubeta$ when the factors of length growing ad infinitum are considered. The basic ingredients of our method are the description of all bispecial factors of $\ubeta$ and the notion of return words. This method can be applied to any fixed point of any primitive substitution.

math.CO↗

arXiv2008

Repetitions in beta-integers

Classical crystals are solid materials containing arbitrarily long periodic repetitions of a single motif. In this paper, we study the maximal possible repetition of the same motif occurring in beta-integers -- one dimensional models of quasicrystals. We are interested in beta-integers realizing only a finite number of distinct distances between neighboring elements. In such a case, the problem may be reformulated in terms of combinatorics on words as a study of the index of infinite words coding beta-integers. We will solve a particular case for beta being a quadratic non-simple Parry number.

math.CO↗

arXiv2008

Palindromic Saturation

We consider two {seemingly} different definitions of infinite words which contain {the} utmost number of palindromes. We show that these two definitions coincide. {The keynote of the proof is a meticulous inspection of properties of complete return words and the application of some basic graph theory.} In fact, we provide another proof of the result announced in \cite{Zamboni}.

math.CO↗

arXiv2008

Asymptotic behavior of beta-integers

Beta-integers (``$β$-integers'') are those numbers which are the counterparts of integers when real numbers are expressed in irrational basis $β> 1$. In quasicrystalline studies $β$-integers supersede the ``crystallographic'' ordinary integers. When the number $β$ is a Parry number, the corresponding $β$-integers realize only a finite number of distances between consecutive elements and somewhat appear like ordinary integers, mainly in an asymptotic sense. In this letter we make precise this asymptotic behavior by proving four theorems concerning Parry $β$-integers.

math-ph↗

arXiv2007

Characterization of substitution invariant 3iet words

We study infinite words coding an orbit under an exchange of three intervals which have full complexity $\C(n)=2n+1$ for all $n\in\N$ (non-degenerate 3iet words). In terms of parameters of the interval exchange and the starting point of the orbit we characterize those 3iet words which are invariant under a primitive substitution. Thus, we generalize the result recently obtained for sturmian words.

math.CO↗

arXiv2007

Matrices of 3iet preserving morphisms

We study matrices of morphisms preserving the family of words coding 3-interval exchange transformations. It is well known that matrices of morphisms preserving sturmian words (i.e. words coding 2-interval exchange transformations with the maximal possible factor complexity) form the monoid $\{\boldsymbol{M}\in\mathbb{N}^{2\times 2} | \det\boldsymbol{M}=\pm1\} = \{\boldsymbol{M}\in\mathbb{N}^{2\times 2} | \boldsymbol{M}\boldsymbol{E}\boldsymbol{M}^T = \pm\boldsymbol{E}\}$, where $\boldsymbol{E} = (\begin{smallmatrix}0&1 -1&0\end{smallmatrix})$. We prove that in case of exchange of three intervals, the matrices preserving words coding these transformations and having the maximal possible subword complexity belong to the monoid $\{\boldsymbol{M}\in\mathbb{N}^{3\times 3} | \boldsymbol{M}\boldsymbol{E}\boldsymbol{M}^T = \pm\boldsymbol{E},\ \det\boldsymbol{M}=\pm 1\}$, where $\boldsymbol{E} = \Big(\begin{smallmatrix}0&1&1 -1&0&1 -1&-1&0\end{smallmatrix}\Big)$.

math.CO↗

arXiv2007

On a class of infinite words with affine factor complexity

In this article, we consider the factor complexity of a fixed point of a primitive substitution canonically defined by a beta-numeration system. We provide a necessary and sufficient condition on the Renyi expansion of 1 for having an affine factor complexity map C(n), that is, such that C(n)=an+b for any integer n.

math.CO↗

arXiv2003

Automorphisms of the fine grading of sl(n,C) associated with the generalized Pauli matrices

We consider the grading of $sl(n,\mathbb{C})$ by the group $Π_n$ of generalized Pauli matrices. The grading decomposes the Lie algebra into $n^2-1$ one--dimensional subspaces. In the article we demonstrate that the normalizer of grading decomposition of $sl(n,\mathbb{C})$ in $Π_n$ is the group $SL(2, \mathbb{Z}_n)$, where $\mathbb{Z}_n$ is the cyclic group of order $n$. As an example we consider $sl(3,\mathbb{C})$ graded by $Π_3$ and all contractions preserving that grading. We show that the set of 48 quadratic equations for grading parameters splits into just two orbits of the normalizer of the grading in $Π_3$.

math-ph↗

arXiv2001

Diophantine equations related to quasicrystals: a note

We give the general solution of three Diophantine equations in the ring of integer of the algebraic number field ${\bf Q}[{\sqr 5}]$. These equations are related to the problem of determination of the minimum distance in quasicrystals with fivefold symmetry.

math-ph↗