On Pansiot Words Avoiding 3-Repetitions

The recently confirmed Dejean's conjecture about the threshold between avoidable and unavoidable powers of words gave rise to interesting and challenging problems on the structure and growth of threshold words. Over any finite alphabet with k >= 5 letters, Pansiot words avoiding 3-repetitions form a regular language, which is a rather small superset of the set of all threshold words. Using cylindric and 2-dimensional words, we prove that, as k approaches infinity, the growth rates of complexity for these regular languages tend to the growth rate of complexity of some ternary 2-dimensional language. The numerical estimate of this growth rate is about 1.2421.Comment: In Proceedings WORDS 2011, arXiv:1108.341

Dejean's conjecture holds for n >= 30

We extend Carpi's results by showing that Dejean's conjecture holds for n >= 30.Comment: introductory material added, minor corrections, 6 page

Binary words containing infinitely many overlaps

We characterize the squares occurring in infinite overlap-free binary words and construct various alpha power-free binary words containing infinitely many overlaps.Comment: 9 page

On the Growth Rates of Complexity of Threshold Languages

Threshold languages, which are the (k/(k-1))+-free languages over k-letter alphabets with k ≥, are the minimal infinite power-free languages according to Dejean's conjecture, which is now proved for all alphabets. We study the growth properties of these languages. On the base of obtained structural properties and computer-assisted studies we conjecture that the growth rate of complexity of the threshold language over k letters tends to a constant α̌ ≈ 1.242 as k tends to infinity. © 2010 EDP Sciences.The authors heartly thank the referees for their valuable comments and remarks

Relations on words

In the first part of this survey, we present classical notions arising in combinatorics on words: growth function of a language, complexity function of an infinite word, pattern avoidance, periodicity and uniform recurrence. Our presentation tries to set up a unified framework with respect to a given binary relation. In the second part, we mainly focus on abelian equivalence, $k$-abelian equivalence, combinatorial coefficients and associated relations, Parikh matrices and $M$-equivalence. In particular, some new refinements of abelian equivalence are introduced

The Weak Circular Repetition Threshold Over Large Alphabets

The repetition threshold for words on $n$ letters, denoted \mbox{RT}(n), is the infimum of the set of all $r$ such that there are arbitrarily long $r$-free words over $n$ letters. A repetition threshold for circular words on $n$ letters can be defined in three natural ways, which gives rise to the weak, intermediate, and strong circular repetition thresholds for $n$ letters, denoted \mbox{CRT}_{\mbox{W}}(n), \mbox{CRT}_{\mbox{I}}(n), and \mbox{CRT}_{\mbox{S}}(n), respectively. Currie and the present authors conjectured that \mbox{CRT}_{\mbox{I}}(n)=\mbox{CRT}_{\mbox{W}}(n)=\mbox{RT}(n) for all $n\geq 4$. We prove that \mbox{CRT}_{\mbox{W}}(n)=\mbox{RT}(n) for all $n\geq 45$, which confirms a weak version of this conjecture for all but finitely many values of $n$.Comment: arXiv admin note: text overlap with arXiv:1911.0577

Avoiding letter patterns in ternary square-free words

We consider special patterns of lengths 5 and 6 in a ternary alphabet. We show that some of them are unavoidable in square-free words and prove avoidability of the other ones. Proving the main results, we use Fibonacci words as codes of ternary words in some natural coding system and show that they can be decoded to square- free words avoiding the required patterns. Furthermore, we estimate the minimal local (critical) exponents of square-free words with such avoidance properties. © 2016, Australian National University. All rights reserved