Edita Pelantova

• PhD
• Professor (Full) at Czech Technical University in Prague

We study a class of infinite words $x_k$ , where $k$ is a positive integer, recently introduced by J. Shallit. This class includes the Thue-Morse sequence $x_1$, the Fibonacci-Thue-Morse sequence $x_2$, and the Allouche-Johnson sequence $x_3$. Shallit stated and for $k = 3$ proved two conjectures on properties of $x_k$. The first conjecture concern...

We consider Cantor real numeration system as a frame in which every non-negative real number has a positional representation. The system is defined using a bi-infinite sequence \(B=(\beta_n)_{n\in\mathbb{Z}}\) of real numbers greater than one. We introduce the set of B-integers and code the sequence of gaps between consecutive B-integers by a symbo...

Using three examples of sequences over a finite alphabet, we want to draw attention to the fact that these sequences having the minimum critical exponent in a given class of sequences show a large degree of symmetry, i.e., they are G-rich with respect to a group G generated by more than one antimorphism. The notion of G-richness generalizes the not...

We consider a numeration system which is a common generalization of the positional systems introduced by Cantor and Rényi. Number representations are obtained using a composition of \(\beta _k\)-transformations for a given sequence of real bases \(\varvec{{\mathcal {B}}}=(\beta _k)_{k\ge 1}\), \(\beta _k>1\). We focus on arithmetical properties of...

We study two positional numeration systems which are known for allowing very efficient addition and multiplication of complex numbers. The first one uses the base $\beta = \imath - 1$ and the digit set $\mathcal{D} = \{ 0, \pm 1, \pm \imath \}$. In this numeration system, every non-zero Gaussian integer~$x$ has an infinite number of representations...

The asymptotic critical exponent measures for a sequence the maximum repetition rate of factors of growing length. The infimum of asymptotic critical exponents of sequences of a certain class is called the asymptotic repetition threshold of that class. On the one hand, if we consider the class of all d-ary sequences with d greater than one, then th...

We define a new class of ternary sequences that are 2-balanced. These sequences are obtained by colouring of Sturmian sequences. We show that the class contains sequences of any given letter frequencies. We provide an upper bound on factor and abelian complexity of these sequences. Using the interpretation by rectangle exchange transformation, we p...

We study the Cantor real base numeration system which is a common generalization of two positional systems, namely the Cantor system with a sequence of integer bases and the Rényi system with one real base. We focus on the case of an alternate base $$\varvec{\mathcal {B}}$$ B given by a purely periodic sequence $$(\beta _n)_{n\ge 1}$$ ( β n ) n ≥ 1...

Fractions $\frac{p}{q} \in [0,1)$ with prime denominator $q$ written in decimal have a curious property described by Midy's Theorem, namely that two halves of their period (if it is of even length $2n$) sum up to $10^n-1$. A number of results generalise Midy's theorem to expansions of $\frac{p}{q}$ in different integer bases, considering non-prime...

We define a new class of ternary sequences that are 2-balanced. These sequencesare obtained by colouring of Sturmian sequences. We show that the class containssequences of any given letter frequencies. We provide an upper bound on factorand abelian complexity of these sequences. Using the interpretation by rectangleexchange transformation, we prove...

The repetition threshold of a class $C$ of infinite $d$-ary sequences is the smallest real number $r$ such that in the class $C$ there exists a sequence that avoids $e$-powers for all $e> r$. This notion was introduced by Dejean in 1972 for the class of all sequences over a $d$-letter alphabet. Thanks to the effort of many authors over more than 30...

To represent real m-dimensional vectors, a positional vector system given by a non-singular matrix M ∈ ℤm×m and a digit set Ɗ ⊂ ℤm is used. If m = 1, the system coincides with the well known numeration system used to represent real numbers. We study some properties of the vector systems which are transformable from the case m = 1 to higher dimensio...

We define a new class of ternary and quaternary sequences that are 2-balanced. These sequences are obtained by colouring of Sturmian sequences. We provide an upper bound on factor and abelian complexity of these sequences. In case of ternary sequences, the factor complexity is at most quadratic, in case of quaternary sequences, at most cubic. We st...

For alternate Cantor real base numeration systems we generalize the result of Frougny and Solomyak on arithmetics on the set of numbers with finite expansion. We provide a class of alternate bases which satisfy the so-called finiteness property. The proof uses rewriting rules on the language of expansions in the corresponding numeration system. The...

To represent real $m$-dimensional vectors, a positional vector system given by a non-singular matrix $M \in \mathbb{Z}^{m \times m}$ and a digit set $\mathcal{D} \subset \mathbb{Z}^m$ is used. If $m = 1$, the system coincides with the well known numeration system used to represent real numbers. We study some properties of the vector systems which a...

For alternate Cantor real base numeration systems we generalize the result of Frougny and~Solomyak on~arithmetics on the set of numbers with finite expansion. We provide a class of alternate bases which satisfy the so-called finiteness property. The proof uses rewriting rules on the~language of~expansions in the corresponding numeration system. The...

The critical exponent E ( u ) E(\mathbf u) of an infinite sequence u \mathbf u over a finite alphabet expresses the maximal repetition of a factor in u \mathbf u . By the famous Dejean’s theorem, E ( u ) ≥ 1 + 1 d − 1 E(\mathbf u) \geq 1+\frac 1{d-1} for every d d -ary sequence u \mathbf u . We define the asymptotic critical exponent E ∗ ( u ) E^*(...

We colour the Fibonacci sequence by suitable constant gap sequences to provide an upper bound on the asymptotic repetitive threshold of $d$-ary balanced sequences. The bound is attained for $d=2, 4$ and $8$ and we conjecture that it happens for infinitely many even $d$'s. Our bound reveals an essential difference in behavior of the repetitive thres...

We study aperiodic balanced sequences over finite alphabets. A sequence v of this type is fully characterised by a Sturmian sequence u and two constant gap sequences y and y′. We show that the language of v is eventually dendric and we focus on return words to its factors. We develop a method for computing the critical exponent and asymptotic criti...

The critical exponent $E(\mathbf u)$ of an infinite sequence $\mathbf u$ over a finite alphabet expresses the maximal repetition of a factor in $\mathbf u$. By the famous Dejean's theorem, $E(\mathbf u) \geq 1+\frac1{d-1}$ for every $d$-ary sequence $\mathbf u$. We define the asymptotic critical exponent $E^*(\mathbf u)$ as the upper limit of the m...

We study the threshold between avoidable and unavoidable repetitions in infinite balanced sequences over finite alphabets. The conjecture stated by Rampersad, Shallit and Vandomme says that the minimal critical exponent of balanced sequences over the alphabet of size d≥5 equals d−2d−3. This conjecture is known to hold for d∈{5,6,7,8,9,10}. We refut...

It is known that each word of length n contains at most n + 1 distinct palindromes. A finite rich word is a word with maximal number of palindromic factors. The definition of palindromic richness can be naturally extended to infinite words. Sturmian words and Rote complementary symmetric sequences form two classes of binary rich words, while epistu...

The set of morphisms mapping any Sturmian sequence to a Sturmian sequence forms together with composition the so-called monoid of Sturm. For this monoid, we defne a faithful representation by $(3\times 3)$-matrices with integer entries. We find three convex cones in $\mathbb{R}^3$ and show that a matrix $R \in Sl(\mathbb{Z},3)$ is a matrix represen...

The first aim of this article is to give information about the algebraic properties of alternate bases $\boldsymbol{\beta}=(\beta_0,\dots,\beta_{p-1})$ determining sofic systems. We show that a necessary condition is that the product $\delta=\prod_{i=0}^{p-1}\beta_i$ is an algebraic integer and all of the bases $\beta_0,\ldots,\beta_{p-1}$ belong t...

We consider an infinite word $\boldsymbol{u}$ fixed by a primitive morphism. We show a necessary condition under which $\boldsymbol{u}$ has a non-trivial geometric representation which is bounded distance equivalent to a lattice.

We study the threshold between avoidable and unavoidable repetitions in infinite balanced sequences over finite alphabets. The conjecture stated by Rampersad, Shallit and Vandomme says that the minimal critical exponent of balanced sequences over the alphabet of size $d \geq 5$ equals $\frac{d-2}{d-3}$. This conjecture is known to hold for $d\in \{...

We show that any m×m matrix M with integer entries and det⁡M=Δ≠0 can be equipped by a finite digit set D⊂Zm such that any integer m-dimensional vector belongs to the setFinD(M)={∑k∈IMkdk:∅≠I finite subset of Z and dk∈D for each k∈I}⊂⋃k∈N1ΔkZm. We also characterize the matrices M for which the sets FinD(M) and ⋃k∈N1ΔkZm coincide.

We study balanced sequences over a d-letter alphabet. Each such sequence \(\mathbf{v}\) is described by a Sturmian sequence and two constant gap sequences \(\mathbf{y}\) and \(\mathbf{y}'\). We provide an algorithm which for a given \(\mathbf{y}\), \(\mathbf{y}'\) and a quadratic slope of a Sturmian sequence computes the critical exponent of the ba...

The Markov numbers are the positive integer solutions of the Diophantine equation x2+y2+z2=3xyz. Already in 1880, Markov showed that all these solutions could be generated along a binary tree. So it became quite usual (and useful) to index the Markov numbers by the rationals from [0,1] which stand at the same place in the Stern–Brocot binary tree....

We study aperiodic balanced sequences over finite alphabets. A sequence v of this type is fully characterised by a Sturmian sequence u and two constant gap sequences y and y'. We show that the language of v is eventually dendric and we focus on return words to its factors. We deduce a method computing critical exponent and asymptotic critical expon...

Cut-and-project sets Σ⊂Rn represent one of the types of uniformly discrete relatively dense sets. They arise by projection of a section of a higher-dimensional lattice to a suitably oriented subspace. Cut-and-project sets find application in solid state physics as mathematical models of atomic positions in quasicrystals, the description of their sy...

We show that any $m\times m$ matrix $M$ with integer entries and $\det M =\Delta \neq 0$ can be equipped by a finite digit set $\mathcal{D}\subset\mathbb{Z}^m$ such that any integer $m$-dimensional vector belongs to the set $$ {\rm Fin}_{\mathcal{D}}(M)= \Bigl\{\sum_{k\in I}M^k {d}_k : \emptyset\neq I \text{ finite subset of } \mathbb{Z} \text{ and...

Occurrences of a factor w in an infinite uniformly recurrent sequence u can be encoded by an infinite sequence over a finite alphabet. This sequence is usually denoted du(w) and called the derived sequence to w in u. If w is a prefix of a fixed point u of a primitive substitution φ, then by Durand's result from 1998, the derived sequence du(w) is f...

We study aperiodic balanced sequences over finite alphabets. A sequence v of this type is fully characterised by a Sturmian sequence u and two constant gap sequences y and y′. We study the language of v, with focus on return words to its factors. We provide a uniform lower bound on the asymptotic critical exponent of all sequences v arising by y an...

Spectra of suitably chosen Pisot-Vijayaraghavan numbers represent non-trivial examples of self-similar Delone point sets of finite local complexity, indispensable in quasicrystal modeling. For the case of quadratic Pisot units we characterize, dependingly on digits in the corresponding numeration systems, the spectra which are bounded distance to a...

The Markov numbers are the positive integer solutions of the Diophantine equation $x^2 + y^2 + z^2 = 3xyz$. Already in 1880, Markov showed that all these solutions could be generated along a binary tree. So it became quite usual (and useful) to index the Markov numbers by the rationals between 0 and 1 which stand at the same place in the Stern-Broc...

Spectra of suitably chosen Pisot-Vijayaraghavan numbers represent non-trivial examples of self-similar Delone point sets of finite local complexity, indispensable in quasicrystal modeling. For the case of quadratic Pisot units we characterize, dependingly on digits in the corresponding numeration systems, the spectra which are bounded distance to a...

We introduce two classes of morphisms over the alphabet A={0,1} whose fixed points contain infinitely many antipalindromic factors. An antipalindrome is a finite word invariant under the action of the antimorphism E:{0,1}∗→{0,1}∗, defined by E(w1⋯wn)=(1−wn)⋯(1−w1). We conjecture that these two classes contain all morphisms (up to conjugation) which...

The non-repetitive complexity nrCu and the initial non-repetitive complexity inrCu are functions which reflect the structure of the infinite word u with respect to the repetitions of factors of a given length. We determine nrCu for the Arnoux–Rauzy words and inrCu for the standard Arnoux–Rauzy words. Our main tools are the S-adic representation of...

Droubay, Justin and Pirillo that each word of length $n$ contains at most $n+1$ distinct palindromes. A finite "rich word" is a word with maximal number of palindromic factors. The definition of palindromic richness can be naturally extended to infinite words. Sturmian words and Rote complementary symmetric sequences form two classes of binary rich...

We determine the critical exponent and the recurrence function of complementary symmetric Rote sequences. The formulae are expressed in terms of the continued fraction expansions associated with the S-adic representations of the corresponding standard Sturmian sequences. The results are based on a thorough study of return words to bispecial factors...

The non-repetitive complexity $nr\mathcal{C}_{\bf u}$ and the initial non-repetitive complexity $inr\mathcal{C}_{\bf u}$ are functions which reflect the structure of the infinite word ${\bf u}$ with respect to the repetitions of factors of a given length. We determine $nr\mathcal{C}_{\bf u}$ for the Arnoux-Rauzy words and $inr\mathcal{C}_{\bf u}$ f...

Cut-and-project sets $\Sigma\subset\mathbb{R}^n$ represent one of the types of uniformly discrete relatively dense sets. They arise by projection of a higher-dimensional lattice to suitably oriented subspaces. Cut-and-project sets find application in solid state physics as mathematical models of atomic positions in quasicrystals, the description of...

Occurrences of a factor $w$ in an infinite uniformly recurrent sequence ${\bf u}$ can be encoded by an infinite sequence over a finite alphabet. This sequence is usually denoted ${\bf d_{\bf u}}(w)$ and called the derived sequence to $w$ in ${\bf u}$. If $w$ is a prefix of a fixed point ${\bf u}$ of a primitive substitution $\varphi$, then by Duran...

Frid, Puzynina and Zamboni (2013) defined the palindromic length of a finite word w as the minimal number of palindromes whose concatenation is equal to w. For an infinite word \(\varvec{u}\) we study \(\mathrm {pal}_{\varvec{u}}\), that is, the function that assigns to each positive integer n, the maximal palindromic length of factors of length n...

Complementary symmetric Rote sequences are binary sequences which have factor complexity C ( n ) = 2 n for all integers n ≥ 1 and whose languages are closed under the exchange of letters. These sequences are intimately linked to Sturmian sequences. Using this connection we investigate the return words and the derived sequences to the prefixes of an...

We introduce two classes of morphisms over the alphabet $A=\{0,1\}$ whose fixed points contain infinitely many antipalindromic factors. An antipalindrome is a finite word invariant under the action of the antimorphism $\mathrm{E}:\{0,1\}^*\to\{0,1\}^*$, defined by $\mathrm{E}(w_1\cdots w_n)=(1-w_{n})\cdots(1-w_1)$. We conjecture that these two clas...

We study the palindromic length of factors of infinite words fixed by morphisms of the so-called class P introduced by Hof, Knill and Simon. We show that it grows at most logarithmically with the length of the factor. For the Fibonacci word and the Thue–Morse word we provide explicit bounds on their rate of growth. We also construct an infinite wor...

Complementary symmetric Rote sequences are binary sequences which have factor complexity $\mathcal{C}(n) = 2n$ for all integers $n \geq 1$ and whose languages are closed under the exchange of letters. These sequences are intimately linked to Sturmian sequences. Using this connection we investigate the return words and the derivated sequences to the...

We study the palindromic length of factors of infinite words fixed by morphisms of the so-called class $\mathcal{P}$ introduced by Hof, Knill and Simon. We show that it grows at most logarithmically with the length of the factor. For the Fibonacci word and the Thue-Morse word we provide estimates on the constants of the growth. We also construct an...

Frid, Puzynina and Zamboni (2013) defined the palindromic length of a finite word $w$ as the minimal number of palindromes whose concatenation is equal to $w$. For an infinite word $u$ we study $PL_{u}$, that is, the function that assigns to each positive integer $n$, the maximal palindromic length of factors of length $n$ in $u$. Recently, Frid (2...

Let the base β be a complex number, | β| > 1 , and let A⊂ C be a finite alphabet of digits. The A-spectrum of β is the set SA(β)={Σk=0nakβk|n∈N,ak∈A}. We show that the spectrum SA(β) has an accumulation point if and only if 0 has a particular (β, A) -representation, said to be rigid. The first application is restricted to the case that β> 1 and the...

Any infinite uniformly recurrent word ${\bf u}$ can be written as concatenation of a finite number of return words to a chosen prefix $w$ of ${\bf u}$. Ordering of the return words to $w$ in this concatenation is coded by derivated word $d_{\bf u}(w)$. In 1998, Durand proved that a fixed point ${\bf u}$ of a primitive morphism has only finitely man...

Any infinite uniformly recurrent word ${\bf u}$ can be written as concatenation of a finite number of return words to a chosen prefix $w$ of ${\bf u}$. Ordering of the return words to $w$ in this concatenation is coded by derivated word $d_{\bf u}(w)$. In 1998, Durand proved that a fixed point ${\bf u}$ of a primitive morphism has only finitely man...

We study periodic expansions in positional number systems with a base $\beta\in\C,\ |\beta|>1$, and with coefficients in a finite set of digits $\A\subset\C.$ We are interested in determining those algebraic bases for which there exists $\A\subset \Q(\beta),$ such that all elements of $\Q(\beta)$ admit at least one eventually periodic representatio...

We study the set of finite words with zero palindromic defect, i.e., words rich in palindromes. This set is factorial, but not recurrent. We focus on description of pairs of rich words which cannot occur simultaneously as factors of a longer rich word.

We study the set of finite words with zero palindromic defect, i.e., words rich in palindromes. This set is factorial, but not recurrent. We focus on description of pairs of rich words which cannot occur simultaneously as factors of a longer rich word.

We study purely morphic words coding symmetric non-degenerate three interval exchange transformation which are known to be palindromic, i.e., they contain infinitely many palindromes. We prove that such words are fixed by a conjugate to a morphism of class , that is, a morphism such that each letter is mapped to where and are both palindromes. We t...

We focus on a generalization of the three gap theorem well known in the framework of exchange of two intervals. For the case of three intervals, our main result provides an analogue of this result implying that there are at most 5 gaps. To derive this result, we give a detailed description of the return times to a subinterval and the corresponding...

A positional numeration system is given by a base and by a set of digits. The base is a real or complex number $\beta$ such that $|\beta|>1$, and the digit set $A$ is a finite set of digits including $0$. Thus a number can be seen as a finite or infinite string of digits. An on-line algorithm processes the input piece-by-piece in a serial fashion....

A positional numeration system is given by a base and by a set of digits. The base is a real or complex number $\beta$ such that $|\beta|>1$, and the digit set $A$ is a finite set of digits including $0$. Thus a number can be seen as a finite or infinite string of digits. An on-line algorithm processes the input piece-by-piece in a serial fashion....

Brlek et al. conjectured in 2008 that any fixed point of a primitive morphism with finite palindromic defect is either periodic or its palindromic defect is zero. Bucci and Vaslet disproved this conjecture in 2012 by a counterexample over ternary alphabet. We prove that the conjecture is valid on binary alphabet. We also describe a class of morphis...

Brlek et al. conjectured in 2008 that any fixed point of a primitive morphism with finite palindromic defect is either periodic or its palindromic defect is zero. Bucci and Vaslet disproved this conjecture in 2012 by a counterexample over ternary alphabet. We prove that the conjecture is valid on binary alphabet. We also describe a class of morphis...

We study periodic expansions in positional number systems with a base $\beta\in\C,\ |\beta|>1$, and with coefficients in a finite set of digits $\A\subset\C.$ We are interested in determining those algebraic bases for which there exists $\A\subset \Q(\beta),$ such that all elements of $\Q(\beta)$ admit at least one eventually periodic representatio...

For a qualitative analysis of spectra of a rectangular analogue of Pais-Uhlenbeck quantum oscillator several rigorous methods of number theory are shown productive and useful. These methods (and, in particular, a generalization of the concept of Markov constant known in Diophantine approximation theory) are shown to provide an entirely new mathemat...

Let $\beta >1 $, $d$ a positive integer, and $$Z_{\beta,d}=\{z_{1} z_{2}\cdots \mid \sum_{i\ge 1}z_i \beta^{-i}=0, \; z_i \in \{-d, \ldots, d\}\}$$ be the set of infinite words having value 0 in base $\beta$ on the alphabet $\{-d, \ldots, d\}$. Based on a recent result of Feng on spectra of numbers, we prove that if the set $Z_{\beta,\lceil \beta \...

For a qualitative analysis of spectra of certain two-dimensional rectangular-well quantum systems several rigorous methods of number theory are shown productive and useful. These methods (and, in particular, a generalization of the concept of Markov constant known in Diophantine approximation theory) are shown to provide a new mathematical insight...

We consider words coding non-degenerate 3 interval exchange transformation. It is known that such words contain infinitely many palindromic factors. We show that for any morphism $\xi$ fixing such a word, either $\xi$ or $\xi^2$ is conjugate to a class $P$ morphism. By this, we provide a new family of palindromic infinite words satisfying the conje...

Fixed points ${\bf u}=\varphi({\bf u})$ of marked and primitive morphisms $\varphi$ over arbitrary alphabet are considered. We show that if ${\bf u}$ is palindromic, i.e., its language contains infinitely many palindromes, then some power of $\varphi$ has a conjugate in class ${\mathcal P}$. This class was introduced by Hof, Knill, Simon (1995) in...

A narrow connection between infinite binary words rich in classical palindromes and infinite binary words rich simultaneously in palindromes and pseudopalindromes (the so-called $H$-rich words) is demonstrated. The correspondence between rich and $H$-rich words is based on the operation $S$ acting over words over the alphabet $\{0,1\}$ and defined...

The m-bonacci word is a generalization of the Fibonacci word to the m-letter alphabet A=0,...,m−1. It is the unique fixed point of the Pisot--type substitution φ:0→01,1→02,...,(m−2)→0(m−1),and(m−1)→0. A result of Adamczewski implies the existence of constants c(m) such that the m-bonacci word is c(m)-balanced, i.e., numbers of letter a occurring in...

The spectrum of a real number $\beta>1$ is the set $X^{m}(\beta)$ of $p(\beta)$ where $p$ ranges over all polynomials with coefficients restricted to ${\mathcal A}=\{0,1,\dots,m\}$. For a quadratic Pisot unit $\beta$, we determine the values of all distances between consecutive points and their corresponding frequencies, by recasting the spectra in...

In 1961 Avizienis proposed a parallel algorithm for addition in base 10 with digit set A = {-6, -5, ..., 5, 6}. Such an algorithm performs addition in constant time, independently of the length of the representation of the summands. In computer arithmetic parallel addition is used for speeding up multiplication and division algorithms. In this work...

Parallel addition in integer base is used for speeding up multiplication and division algorithms. $k$-block parallel addition has been introduced by Kornerup in 1999: instead of manipulating single digits, one works with blocks of fixed length $k$. The aim of this paper is to investigate how such notion influences the relationship between the base...

For real $q > 1$, Erdos, Joo and Komornik study distances of the consecutive points in the set $X^m(q)=\{a_0+a_1q+\cdots+a_nq^n : n\geq0,\ a_i\in\{0,\ldots,m\}\}$. The Pisot numbers play a crucial role for properties of $X^m(q)$. We follow work of Zaimi who consideres $X^m(\gamma)$ with a non-real $\gamma$ and $|\gamma| > 1$. We show that for any n...

We focus on the exchange T of two intervals with an irrational slope α. For a general subinterval I of the domain of T, the first return time to I takes three values. We describe the structure of the set of return itineraries to I. In particular, we show that it is equal to {R1,R2,R1R2,Q} where Q is amicable with R{1, R2 or R1R2.

Generalized pseudostandard word $\bf u$, as introduced in 2006 by de Luca and De Luca, is given by a directive sequence of letters from an alphabet ${\cal A}$ and by a directive sequence of involutory antimorphisms acting on ${\cal A}^*$. Prefixes of $\bf u$ with increasing length are constructed using pseudopalindromic closure operator. We show th...

Generalized pseudostandard word $\bf u$, as introduced in 2006 by de Luca and De Luca, is given by a directive sequence of letters from an alphabet ${\cal A}$ and by a directive sequence of involutory antimorphisms acting on ${\cal A}^*$. Prefixes of $\bf u$ with increasing length are constructed using pseudopalindromic closure operator. We show th...

We follow the works of Puzynina and Zamboni, and Rigo et al. on abelian returns in Sturmian words. We determine the cardinality of the set $\mathcal{APR}_u$ of abelian returns of all prefixes of a Sturmian word $u$ in terms of the coefficients of the continued fraction of the slope, dependingly on the intercept. We provide a simple algorithm for fi...

We study parallel algorithms for addition of numbers having finite representation in a positional numeration system defined by a base β in ℂ and a finite digit set A of contiguous integers containing 0. For a fixed base β, we focus on the question of the size of the alphabet that permits addition in constant time, independently of the length of rep...

Brlek and Reutenauer conjectured that any infinite word u with language closed under reversal satisfies the equality 2D(u)=∑n=0+∞Tu(n) in which D(u) denotes the defect of u and Tu(n) denotes Cu(n+1)−Cu(n)+2−Pu(n+1)−Pu(n), where Cu and Pu are the factor and palindromic complexity of u, respectively. This conjecture was verified for periodic words by...

Bašić (2012) in [1] pointed to a gap in the proof of Corollary 5.10 in Balková et al. (2011) [2] related to the Brlek–Reutenauer conjecture. In this corrigendum, we correct the proof and show that the corollary remains valid.

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 $\Z_{-\beta}$ of numbers whose representation uses only non-negative powers of $-\beta$, the so-called $(-\beta)$-integers. We describe the distances between consecutive elements of...

We focus on Θ-rich and almost Θ-rich words over a finite alphabet $\mathcal{A}$, where Θ is an involutive antimorphism over $\mathcal{A}^{\ast}$. We show that any recurrent almost Θ-rich word u is an image of a recurrent Θ′-rich word under a suitable morphism, where Θ′ is also an involutive antimorphism. Moreover, if the word u is uniformly recurre...

Brlek and Reutenauer conjectured that any infinite word u with language closed under reversal satisfies the equality 2D(u) = \sum_{n=0}^{\infty}T_u(n) in which D(u) denotes the defect of u and T_u(n) denotes C_u(n+1)-C_u(n) +2 - P_U(n+1) - P_u(n), where C_u and P_u are the factor and palindromic complexity of u, respectively. This conjecture was ve...

For a given base $\gamma$ and a digit set ${\mathcal B}$ we consider optimal representations of a number $x$, as defined by Dajani at al. in 2012. For a non-integer negative base $\gamma=-\beta<-1$ and the digit set ${\mathcal A}_\beta:={0,1,...,\lceil\beta\rceil-1}$ we derive the transformation which generates the optimal representation, if it exi...

We study the question of pure periodicity of expansions in the negative base numeration system. In analogy of Akiyama's result for positive Pisot unit base $\beta$, we find a sufficient condition so that there exist an interval $J$ containing the origin such that the $(-\beta)$-expansion of every rational number from $J$ is purely periodic. We focu...

We study non-standard number systems with negative base -beta. Instead of the Ito-Sadahiro definition, based on the transformation T-beta of the interval [-beta/beta+1, 1/beta+1) into itself, we suggest a generalization using an interval [l, l + 1) with l is an element of (-1, 0]. Such numeration systems share many properties of positive base numer...

Graph Theory International audience We consider exchange of three intervals with permutation (3, 2, 1). The aim of this paper is to count the cardinality of the set 3iet (N) of all words of length N which appear as factors in infinite words coding such transformations. We use the strong relation of 3iet words and words coding exchange of two interv...

We consider positional numeration systems with negative real base $-\beta$, where $\beta>1$, and study the extremal representations in these systems, called here the greedy and lazy representations. We give algorithms for determination of minimal and maximal $(-\beta)$-representation with respect to the alternate order. We also show that both extre...

Brlek and Reutenauer conjectured that any infinite word u with language closed under reversal satisfies the equality 2D(u)=\sum_{n=0}^{\infty} T(n) in which D(u) denotes the defect of u and T(n) denotes C(n+1)-C(n)+2-P(n+1)-P(n), where C and P are the factor and palindromic complexity of u, respectively. Brlek and Reutenauer verified their conjectu...

We study repetitions in infinite words coding exchange of three intervals with permutation (3, 2, 1), called 3iet words. The language of such words is determined by two parameters, ϵ,ℓ. We show that finiteness of the index of 3iet words is equivalent to boundedness of the coefficients of the continued fraction of ϵ. In this case, we also give an up...