Lubomira Balkova

Infinite Words with Well Distributed Occurrences

In this paper we introduce the well distributed occurrences WDO combinatorial property for infinite words, which guarantees good behavior no lattice structure in some related pseudorandom number generators. An infinite word u on a d-ary alphabet has the WDO property if, for each factor w of u, positive integer m, and vector v

Palindromic complexity of infinite words associated with non-simple Parry numbers

\in{\mathbb Z}_{m}^{d}

Corrigendum: “On Brlek-Reutenauer conjecture”

, there is an occurrence of w such that the Parikh vector of the prefix of u preceding such occurrence is congruent to v modulo m. We prove that Sturmian words, and more generally Arnoux-Rauzy words and some morphic images of them, have the WDO property.

Factor Frequencies in Languages Invariant Under Symmetries Preserving Factor Frequencies

We study the palindromic complexity of infinite words u β , the fixed points of the substitution over a binary alphabet, φ(0) = 0 a 1 , φ(1) = 0 b 1 , with a - 1 ≥ b ≥ 1 , which are canonically associated with quadratic non-simple Parry numbers β .

Return Words and Recurrence Function of a Class of Infinite Words

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

Continued Fractions of Square Roots of Natural Numbers

Abstract. The number of frequencies of factors of length in a recurrent aperiodic infinite word does not exceed , where is the first difference of factor complexity, as shown by Boshernitzan. Pelantová together with the author derived a better upper bound for infinite words whose language is closed under reversal. In this paper, we further diminish the upper bound for uniformly recurrent infinite words whose language is invariant under all elements of a finite group of symmetries and we prove the optimality of the obtained upper bound.

The Meyer property of cut-and-project sets

Many combinatorial and arithmetical properties have been studied for infinite words ub associated with s-integers. Here, new results describing return words and recurrence function for a special case of ub will be presented. The methods used here can be applied to more general infinite words, but the description then becomes rather technical.

Sturmian jungle (or garden?) on multiliteral alphabets

In this paper, we will first summarize known results concerning continued fractions. Then we will limit our consideration to continued fractions of quadratic numbers. The second author describes periods and sometimes the precise form of continued fractions of ?N, where N is a natural number. In cases where we have been able to find such results in the literature, we recall the original authors, however many results seem to be new.

Proof of the Brlek-Reutenauer conjecture

We consider cut-and-project sets Σ(Ω) with compact acceptance window Ω C R d . It is known that Σ(Ω) satisfies the Meyer property, i.e. it is a Delone set and there exists a finite set F such that Σ(Ω) - Σ(Ω) C Σ(Ω) + F. The investigation of the set F can be transformed into the problem of covering of the difference set Ω - Ω by open copies Ω°. The cardinality f(Ω) of the minimal covering is called the Meyer number of Ω. We study topological properties of the function f and show that it is bounded on the space of convex compact sets Ω ⊂ E d . We give estimates on the universal upper bound of the Meyer number of Ω C R 2 . We further show that f is not bounded if we relax the condition of convexity.