5 research outputs found
Polynomial versus Exponential Growth in Repetition-Free Binary Words
It is known that the number of overlap-free binary words of length n grows
polynomially, while the number of cubefree binary words grows exponentially. We
show that the dividing line between polynomial and exponential growth is 7/3.
More precisely, there are only polynomially many binary words of length n that
avoid 7/3-powers, but there are exponentially many binary words of length n
that avoid (7/3+)-powers. This answers an open question of Kobayashi from 1986.Comment: 12 page
Shortest Repetition-Free Words Accepted by Automata
We consider the following problem: given that a finite automaton of
states accepts at least one -power-free (resp., overlap-free) word, what is
the length of the shortest such word accepted? We give upper and lower bounds
which, unfortunately, are widely separated.Comment: 12 pages, conference pape
The First-Order Theory of Binary Overlap-Free Words is Decidable
We show that the first-order logical theory of the binary overlap-free words
(and, more generally, the -free words for rational , ), is decidable. As a consequence, many results previously
obtained about this class through tedious case- based proofs can now be proved
"automatically", using a decision procedure
The repetition threshold for binary rich words
A word of length is rich if it contains nonempty palindromic factors.
An infinite word is rich if all of its finite factors are rich. Baranwal and
Shallit produced an infinite binary rich word with critical exponent
() and conjectured that this was the least
possible critical exponent for infinite binary rich words (i.e., that the
repetition threshold for binary rich words is ). In this article,
we give a structure theorem for infinite binary rich words that avoid
-powers (i.e., repetitions with exponent at least 2.8). As a consequence,
we deduce that the repetition threshold for binary rich words is
, as conjectured by Baranwal and Shallit. This resolves an open
problem of Vesti for the binary alphabet; the problem remains open for larger
alphabets.Comment: 16 page
Répétitions dans les mots et seuils d'évitabilité
Nous étudions dans cette thèse différents problèmes d'évitabilité des répétitions dans les mots infinis. Soulevée par Thue et motivée par ses travaux sur les mots sans carrés, la problématique s'est développée au cours du XXe siècle, et est aujourd'hui devenue un des grands domaines de recherche en combinatoire des mots. En 1972, Dejean proposa une importante conjecture, dont la validation étape par étape s'est terminée récemment (2009). La conjecture concerne le seuil des répétitions d'un alphabet, i.e., la borne inférieure des exposants évitables sur cet alphabet. La notion de seuil, comme frontière entre évitabilité et non-évitabilité d'un ensemble donné de mots, est le fil directeur de nos travaux. Nous nous intéressons d'abord à une généralisation du seuil des répétitions (nous donnons des encadrements de sa valeur). Cette notion permet d'ajouter, pour décrire l'ensemble des répétitions à éviter, au paramètre de l'exposant, celui de la longueur des répétitions. Puis, nous étudions des problèmes d'existence de mots dans lesquels, simultanément, certaines répétitions sont interdites et d'autres sont forcées. Nous répondons, pour l'alphabet ternaire, à la question : quels réels sont l'exposant critique d'un mot infini sur un alphabet fixé? Nous introduisons ensuite une notion de haute répétitivité, et établissons une description partielle des couples d'exposants paramètrant une double contrainte de haute répétitivité et d'évitabilité. Pour finir, nous utilisons des résultats et techniques issus de ces problématiques pour résoudre une question de coloration de graphes : nous introduisons un seuil des répétitions, calqué sur celui connu pour les mots, et donnons sa valeur pour deux classes de graphes, les arbres et les graphes de subdivisions.In this thesis we study various problems on repetition avoidance in infinite words. Raised by Thue and motivated by his work on squarefree words, the topic developed during the 20th century, and has nowadays become a principal area of research in combinatorics on words. In 1972, Dejean proposed an important conjecture whose verification in steps was completed recently (2009). The conjecture concerns the repetition threshold for an alphabet, i.e., the infimum of the avoidable exponents for that alphabet. The notion of threshold as a borderline between avoidability and unavoidability for a given set of words is the guiding line of our work. First, we focus on a generalization of the repetition threshold. This concept allows us to include, in addition to the exponent, the length of the repetitions as a parameter in the description of the set of repetitions to avoid. We obtain various bounds in that respect. We then study existence problems for words in which simultaneously some repetitions are forbidden, and others are forced. For the ternary alphabet, we answer the question: what real numbers are the critical exponent of some infinite word over a given alphabet? Also, we introduce a notion of highly repetitive words and give a partial description of the pairs of exponents which parameterize the existence of words both highly repetitive and repetition-free. Finally, we use results and techniques stemming from those problems to solve a question on graph colouring: we introduce a repetition threshold adapted from the thresholds we know for words, and give its value for two classes of graphs, namely, trees and subdivision graphs.AIX-MARSEILLE2-Bib.electronique (130559901) / SudocSudocFranceF