Uniformly balanced words with linear complexity and prescribed letter frequencies

Abstract

We consider the following problem. Let us fix a finite alphabet A; for any given d-uple of letter frequencies, how to construct an infinite word u over the alphabet A satisfying the following conditions: u has linear complexity function, u is uniformly balanced, the letter frequencies in u are given by the given d-uple. This paper investigates a construction method for such words based on the use of mixed multidimensional continued fraction algorithms

On the structure of bispecial Sturmian words

“…Given an integer C > 0, a word is said to be C-balanced if for any pair of factors of the same length u and v, and for any letter a, one has ||u| a − |v| a | ≤ C. So what we referred to in this paper as balanced words are precisely the 1-balanced words. V. Berthé and S. Labbé raised the question whether it is possible to construct C-balanced words with linear factor complexity that have prescribed letter frequencies [9]. The same authors also presented interesting results about the algorithmic generation of digital approximations of segments in the 3-dimensional space [8].…”

“…Lower Christoffel word w p,q Upper Christoffel word w p,q (11,1) aaaaaaaaaaab baaaaaaaaaaa (10, 2) aaaaabaaaaab baaaaabaaaaa (9,3) aaabaaabaaab baaabaaabaaa (8,4) aabaabaabaab baabaabaabaa (7,5) aababaababab bababaababaa (6,6) abababababab babababababa (5,7) abababbababb bbababbababa (4,8) abbabbabbabb bbabbabbabba (3,9) abbbabbbabbb bbbabbbabbba (2, 10) abbbbbabbbbb bbbbbabbbbba (1,11) abbbbbbbbbbb bbbbbbbbbbba Note that given a word w in a factorial and extendible language L over an alphabet Σ, if w is a maximal internal factor of a minimal forbidden word for L, then w is a bispecial word in L. In fact, let x, y ∈ Σ be such that xwy is a minimal forbidden word for L. By the definition of minimal forbidden word, xw and wy belong to L. Since L is extendible, there is a letter y = y in Σ such that xwy ∈ L; symmetrically, there is a letter x = x ∈ Σ such that x wy ∈ L. Since L is factorial, wy, wy , xw and x w belong to L, and therefore w is bispecial in L. However, the converse is not true: if w is a strictly bispecial word in L, then it cannot be the maximal internal factor of a minimal forbidden word for L.…”

Factor complexity of S-adic words generated by the Arnoux–Rauzy–Poincaré algorithm

“…The experimentations described in [BL11,Lab12] indicate that some multidimensional continued fraction algorithms generate S-adic words having a linear factor complexity and a bounded balance for almost every letter frequencies vector. In particular, Brun multidimensional continued fraction algorithm as well as the Arnoux-Rauzy-Poincaré algorithm seem to be the two best choices in terms of balance properties.…”

Convergence and Factor Complexity for the Arnoux-Rauzy-Poincaré Algorithm

“…The Arnoux-Rauzy-Poincaré multidimensional continued fraction algorithm is a fusion algorithm such as introduced in [BL11,Lab12]. It is defined on the 2-simplex…”

“…In [BL11,Lab12], we considered this question under the approach of multidimensional continued fraction algorithms and S-adic systems. Experimentations suggested that Brun multidimensional continued fraction algorithm as well as a fusion of Arnoux-Rauzy and Poincaré algorithms were the two best choices to investigate for such an approach.…”