arXiv subjects

Edita Pelantová

Publications and source records attributed to Edita Pelantová.

At least 19 recordsLinked to original sources

arXiv2026

Generalized reflection complexity

The reflection complexity of a sequence, introduced recently by Allouche, Campbell, Li, Shallit, and Stipulanti, counts the factors of a given length up to reversal. We generalize this notion considering a group $G$ generated by antimorphisms over a given alphabet $\mathcal A$. The $G$-complexity of a sequence over $\mathcal A$ counts the factors of a given length up to the action of $G$. For $G$ generated by the reflection alone, the $G$-complexity coincides with the reflection complexity. We characterize eventual periodicity in terms of the $G$-complexity. Then we focus on the binary alphabet and the group $H$ generated by reflection and the antimorphism exchanging letters. We provide a simple formula for the $H$-complexity in the case of $H$-rich sequences and determine the precise values for complementary symmetric Rote sequences and generalized binary Thue-Morse sequences. Finally, we describe the aperiodic recurrent sequences having the minimum $H$-complexity -- they are exactly the complementary symmetric Rote sequences, while the binary sequences with the minimum reflection complexity correspond to the Sturmian sequences.

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arXiv2026

Non-Expansive Matrix Based number Systems

An integer $d$-dimensional vector is usually represented by $d$ strings, which express the value of each component separately. This article is devoted to number systems that allow representing integer vectors using a single string of digits. This system is given by an integer square matrix $M$ and a finite set $\mathcal{D}$ of symbols, called digits. We focus on full number systems, in which every vector $\vec{x} \in \mathbb{Z}^n$ can be written in the form $\vec{x}=\sum_{i=0}^{k-1} M^i d_i$ with $d_i \in \mathcal{D}$ and thus represented by the string $d_{k-1}d_{k-2}\cdots d_1d_0$. In contrast to the very well-described case where the base $M$ is expansive, we focus on systems in which $M$ is similar to a Jordan block with the eigenvalue $\pm 1$. We follow the work of Caldwell, Hare, and Vávra who initiated the study of such systems. First, we address the question of choosing a minimal sized digit set $\mathcal{D}$. Then we describe optimal representations for vectors from $\mathbb{Z}^2$, both with respect to their length and the occurrence of non-zero digits.

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arXiv2026

Parry condition, existence and uniqueness of alternate bases

Alternate bases are a numeration system that generalizes the Rényi numeration system. It is common in this context to construct examples or counter-examples by specifying the expansions of $1$ in the desired system. While it is easy to show when a system with given expansions of $1$ exists in the Rényi case, the same is not true in the alternate case. In this article, we establish conditions for given words to be the expansions of $1$ in the alternate case. To do so, we use a fixed point theorem on matrices defined from the expansions and obtain the elements of the base from the components of the fixed point. We also obtain a partial result for the uniqueness of such a base. In the latter parts of the article, we use similar techniques to prove the existence of bases with a given sequence of $B$-integers.

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arXiv2026

Matrices with cyclically monotone rows and Cantor numeration systems

We study a class of square matrices with non-negative elements which have cyclically monotone rows in the sense that each row of a matrix from the class consists of a cyclically non-increasing sequence of numbers starting from a maximal element on the diagonal. We prove that if every diagonal element is strictly larger than all other elements in the respective row, then the matrix is regular. This property enables us to solve an open problem that comes from the theory of non-standard numeration systems, also called Cantor numeration systems. The problem concerns a one-to-one relationship between Cantor real bases, which are supposed to be alternate, that is, periodic with a period p, and lists of p sequences of non-negative integers satisfying the so-called Parry condition.

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arXiv2026

On a sequence of Kimberling and its relationship to the Tribonacci word

In 2017, Clark Kimberling defined an interesting sequence ${\bf B} = 0100101100 \cdots$ of $0$'s and $1$'s by certain inflation rules, and he made a number of conjectures about this sequence and some related ones. In this note we prove his conjectures using, in part, the Walnut theorem-prover. We show how his word is related to the infinite Tribonacci word, and we determine both the subword complexity and critical exponent of $\bf B$.

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arXiv2025

Reflection on the reflection complexity

The factor complexity ${\mathcal C}_{\mathbf u}$ of a sequence ${\mathbf u} = u_0u_1u_2 \cdots$ over a finite alphabet counts the number of factors of length $n$ occurring in $\mathbf u$, i.e., ${\mathcal C}_{\mathbf u}(n) = \#{\mathcal L}_n(\mathbf u)$, where ${\mathcal L}_n({\mathbf u)}= \{u_iu_{i+1}\cdots u_{i+n-1}: i \in \mathbb N\}$. Two factors of ${\mathcal L}_n(\mathbf u)$ are said to be equivalent if one factor is the reversal of the other one. Recently, Allouche et al. introduced the reflection complexity $r_{\mathbf u}$ which counts the number of non-equivalent factors of $\mathcal{L}_n(\mathbf u)$. They formulated the following conjecture: a sequence $\mathbf u$ is eventually periodic if and only if $r_{\mathbf u}(n+2) = r_{\mathbf u}(n)$ for some $n \in \mathbb N$. Here we prove the conjecture and characterize the sequences for which $r_{\mathbf u}(n+2) = r_{\mathbf u}(n)+1$ for every $n \in \mathbb N$ and also the sequences for which the equality is satisfied for every sufficiently large $n \in \mathbb N$.

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arXiv2025

On two conjectures of Shallit about Thue-Morse-like sequences

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 concerns the factor complexity, the second one the critical exponent of these words. We confirm the validity of both conjectures for every $k$.

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arXiv2025

The limit of repetition thresholds of rich sequences

The repetition threshold of a class of sequences is the smallest number $r$ such that a sequence from the class contains no repetition with exponent $> r$. We focus on the class $\mathcal{C}_d$ of $d$-ary sequences rich in palindromes. In 2020, Currie, Mol, and Rampersad determined the repetition threshold for $\mathcal{C}_2$. In 2024, Currie, Mol and Peltomäki found the repetition threshold for $\mathcal{C}_3$ and conjectured that the repetition threshold for $\mathcal{C}_d$ tends to 2 with $d$ growing to infinity. Here we verify their conjecture.

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arXiv2025

Frequencies of letters in infinite $k$-balanced sequences

Frequency of letters in a symbolic sequence ${\bf u}$ over a finite alphabet is one of the basic characteristics of ${\bf u}$. The notion of $k$-balancedness captures the property that the number of any letter occurring in two arbitrary factors of ${\bf u}$ of equal length differs at most by $k$. For a fixed integer $k$ and alphabet size $d\in \mathbb N$, we discuss possible frequencies of letters in $k$-balanced $d$-ary sequences. For the size $d$ of the alphabet, we introduce the notion of balancedness threshold $BT(d)$ and give an upper bound on it, where $BT(d)$ is the minimum $k$ such that there exists a $k$-balanced sequence over a $d$-letter alphabet for all possible letter frequencies.

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arXiv2025

A note on symmetries of rich sequences with minimum critical exponent

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 notion of richness in palindromes which is based on one antimorphism, namely the reversal mapping. The three examples are: 1) the Thue-Morse sequence which has the minimum critical exponent among all binary sequences; 2) the sequence which has the minimum critical exponent among all binary rich sequences; 3) the sequence which has the minimum critical exponent among all ternary rich sequences.

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arXiv2024

Optimal Representations of Gaussian and Eisenstein Integers using digit sets closed under multiplication

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 $β= \imath - 1$ 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. We focus on optimal representations of~$x$ -- i.e., representations with minimal possible number of non-zero digits. One of the optimal representations of~$x$ has the so-called $3$-non-adjacent form ($3$-NAF). We provide an upper bound on the number of distinct optimal representations of~$x$, depending on the number of non-zero digits in the $3$-NAF of~$x$. We also characterize the Gaussian integers for which the upper bound is attained. The same questions are answered also for the second numeration system with base $β= ω- 1$ and digit set $\mathcal{D} = \{ 0, \pm 1, \pm ω, \pm ω^2 \}$, where $ω= \exp(2π\imath / 3)$. In this system, every Eisenstein integer has a $2$-NAF, which is optimal. This paper can be understood as an analogy to the result of Grabner and Heuberger obtained for the signed binary numeration system, using base $β= 2$ and digit set $\mathcal{D} = \{0, \pm 1\}$.

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arXiv2024

The asymptotic repetition threshold of sequences rich in palindromes

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 the asymptotic repetition threshold is equal to one, independently of the alphabet size. On the other hand, for the class of episturmian sequences, the repetition threshold depends on the alphabet size. We focus on rich sequences, i.e., sequences whose factors contain the maximum possible number of distinct palindromes. The class of episturmian sequences forms a subclass of rich sequences. We prove that the asymptotic repetition threshold for the class of rich recurrent d-ary sequences, with d greater than one, is equal to two, independently of the alphabet size.

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arXiv2024

Midy's Theorem in non-integer bases and divisibility of Fibonacci numbers

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 denominators, or dividing the period into more than two parts. We show that a similar phenomena can be studied even in the context of numeration systems with non-integer bases, as introduced by Rényi. First we define the Midy property for a general real base $β>1$ and derive a necessary condition for validity of the Midy property. For $β=\frac12(1+\sqrt5)$ we characterize prime denominators $q$, which satisfy the property.

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arXiv2024

2-balanced sequences coding rectangle exchange transformation

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 prove that for almost all triples of letter frequencies, the upper bound on factor and abelian complexity is reached. The bound on factor complexity is given using a number-theoretical function which we compute explicitly for a class of parameters.

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arXiv2024

Periodicity and pure periodicity in alternate base systems

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 so-called alternate base $B$ given by a purely periodic sequence of real numbers greater than 1. We answer an open question of Charlier et al. on the set of numbers with eventually periodic $B$-expansions. We also investigate for which bases all sufficiently small rationals have a purely periodic $B$-expansion.

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arXiv2024

Finiteness property in Cantor real numeration systems

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 proof is constructive and provides a~method for~performing addition of~expansions in Cantor real bases. 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 $β_k$-transformations for a given sequence of real bases $B=(β_k)_{k\geq 1}$, $β_k>1$. We focus on~arithmetical properties of the set of numbers with finite $B$-expansion in case that $B$ is an alternate base, i.e.\ $B$ is a periodic sequence. We provide necessary conditions for the so-called finiteness property. We further show a~sufficient condition using rewriting rules on the~language of~representations. The proof is constructive and provides a~method for~performing addition of~expansions in alternate bases. Finally, we give a family of alternate bases that satisfy this sufficient condition. Our work generalizes the results of Frougny and Solomyak obtained for the case when the base $B$ is a constant sequence.

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arXiv2023

Substitutions and Cantor real numeration systems

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 $\Beta=(β_n)_{n\in\Z}$ of real numbers greater than one. We introduce the set of $\Beta$-integers and code the sequence of gaps between consecutive $\Beta$-integers by a symbolic sequence in general over the alphabet $\N$. We show that this sequence is $S$-adic. We focus on alternate base systems, where the sequence $\Beta$ of bases is periodic and characterize alternate bases $\Beta$, in which $\Beta$-integers can be coded using a symbolic sequence $v_{\Beta}$ over a finite alphabet. With these so-called Parry alternate bases we associate some substitutions and show that $v_\Beta$ is a fixed point of their composition. The paper generalizes results of Fabre and Burdík et al.\ obtained for the Rényi numerations systems, i.e., in the case when the Cantor base $\Beta$ is a constant sequence.

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arXiv2023

The repetition threshold of episturmian sequences

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 years, the precise value of the repetition threshold in this class is known for every $d \in \mathbb N$. The repetition threshold for the class of Sturmian sequences was determined by Carpi and de Luca in 2000. Sturmian sequences may be equivalently defined in various ways, therefore there exist many generalizations to larger alphabets. Rampersad, Shallit and Vandome in 2020 initiated a study of the repetition threshold for the class of balanced sequences -- one of the possible generalizations of Sturmian sequences. Here, we focus on the class of $d$-ary episturmian sequences -- another generalization of Sturmian sequences introduced by Droubay, Justin and Pirillo in 2001. We show that the repetition threshold of this class is reached by the $d$-bonacci sequence and its value equals $2+\frac{1}{t-1}$, where $t>1$ is the unique positive root of the polynomial $x^d-x^{d-1}- \cdots -x-1$.

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