arXiv subjects

Lubomíra Dvořáková

Publications and source records attributed to Lubomíra Dvořáková.

17 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

Attractors of sequences coding $β$-integers

In this paper, we describe minimal string attractors of prefixes of simple Parry sequences. These sequences form a coding of distances between consecutive $β$-integers in numeration systems with a real base $β$. Simple Parry sequences have been recently studied from this point of view and attractors of prefixes have been described. However, the authors themselves had doubts about their minimality and conjectured that attractors of alphabet size should be sufficient. We confirm their conjecture. Moreover, we provide minimal attractors of prefixes of some particular form of binary non-simple Parry sequences.

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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

String attractors of Rote sequences

In this paper, we describe minimal string attractors (of size two) of pseudopalindromic prefixes of standard complementary-symmetric Rote sequences. Such a class of Rote sequences forms a subclass of binary generalized pseudostandard sequences, i.e., of sequences obtained when iterating palindromic and antipalindromic closures. When iterating only palindromic closure, palindromic prefixes of standard Sturmian sequences are obtained and their string attractors are of size two. However, already when iterating only antipalindromic closure, antipalindromic prefixes of binary pseudostandard sequences are obtained and we prove that the minimal string attractors are of size three in this case. We conjecture that the pseudopalindromic prefixes of any binary generalized pseudostandard sequence have a minimal string attractor of size at most four.

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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

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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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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arXiv2022

An upper bound on asymptotic repetitive threshold of balanced sequences via colouring of the Fibonacci sequence

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 threshold and the asymptotic repetitive threshold of balanced sequences. The repetitive threshold of $d$-ary balanced sequences is known to be at least $1+\frac{1}{d-2}$ for each $d \geq 3$. In contrast, our bound implies that the asymptotic repetitive threshold of $d$-ary balanced sequences is at most $1+\frac{τ^3}{2^{d-3}}$ for each $d\geq 2$, where $τ$ is the golden mean.

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arXiv2022

String attractors of episturmian sequences

In this paper, we describe string attractors of all factors of episturmian sequences and show that their size is equal to the number of distinct letters contained in the factor.

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arXiv2022

Asymptotic repetitive threshold of balanced sequences

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 maximal repetition of factors of length $n$. We show that for any $d>1$ there exists a $d$-ary sequence $\mathbf u$ having $E^*(\mathbf u)$ arbitrarily close to $1$. Then we focus on the class of $d$-ary balanced sequences. In this class, the values $E^*(\mathbf u)$ are bounded from below by a threshold strictly bigger than 1. We provide a method which enables us to find a $d$-ary balanced sequence with the least asymptotic critical exponent for $2\leq d\leq 10$.

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arXiv2021

On minimal critical exponent of balanced sequences

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 $d\in \{5, 6, 7,8,9,10\}$. We refute this conjecture by showing that the picture is different for bigger alphabets. We prove that critical exponents of balanced sequences over an alphabet of size $d\geq 11$ are lower bounded by $\frac{d-1}{d-2}$ and this bound is attained for all even numbers $d\geq 12$. According to this result, we conjecture that the least critical exponent of a balanced sequence over $d$ letters is $\frac{d-1}{d-2}$ for all $d\geq 11$.

math.CO↗

arXiv2020

Complementary symmetric Rote sequences: the critical exponent and the recurrence function

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 of Sturmian sequences. Using the formula for the critical exponent, we describe all complementary symmetric Rote sequences with the critical exponent less than or equal to 3, and we show that there are uncountably many complementary symmetric Rote sequences with the critical exponent less than the critical exponent of the Fibonacci sequence. Our study is motivated by a~conjecture on sequences rich in palindromes formulated by Baranwal and Shallit. Its recent solution by Curie, Mol, and Rampersad uses two particular complementary symmetric Rote sequences.

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