Transition Property for α\alpha-Power Free Languages with α≥2\alpha\geq 2 and k≥3k\geq 3 Letters

In 1985, Restivo and Salemi presented a list of five problems concerning power free languages. Problem $4$ states: Given $\alpha$-power-free words $u$ and $v$, decide whether there is a transition from $u$ to $v$. Problem $5$ states: Given $\alpha$-power-free words $u$ and $v$, find a transition word $w$, if it exists. Let $\Sigma_k$ denote an alphabet with $k$ letters. Let $L_{k,\alpha}$ denote the $\alpha$-power free language over the alphabet $\Sigma_k$, where $\alpha$ is a rational number or a rational "number with $+$". If $\alpha$ is a "number with $+$" then suppose $k\geq 3$ and $\alpha\geq 2$. If $\alpha$ is "only" a number then suppose $k=3$ and $\alpha>2$ or $k>3$ and $\alpha\geq 2$. We show that: If $u\in L_{k,\alpha}$ is a right extendable word in $L_{k,\alpha}$ and $v\in L_{k,\alpha}$ is a left extendable word in $L_{k,\alpha}$ then there is a (transition) word $w$ such that $uwv\in L_{k,\alpha}$. We also show a construction of the word $w$

Avoiding approximate repetitions with respect to the longest common subsequence distance

Ochem, Rampersad, and Shallit gave various examples of infinite words avoiding what they called approximate repetitions. An approximate repetition is a factor of the form x x', where x and x' are close to being identical. In their work, they measured the similarity of x and x' using either the Hamming distance or the edit distance. In this paper, we show the existence of words avoiding approximate repetitions, where the measure of similarity between adjacent factors is based on the length of the longest common subsequence. Our principal technique is the so-called “entropy compression” method, which has its origins in Moser and Tardos’s algorithmic version of the Lovász local lemma.Rampersad is supported by an NSERC Discovery Grant.https://msp.org/involve/2016/9-4/p07.xhtm

Pattern avoidance: themes and variations

AbstractWe review results concerning words avoiding powers, abelian powers or patterns. In addition we collect/pose a large number of open problems

Extensions and reductions of square-free words

A word is square-free if it does not contain a nonempty word of the form $XX$ as a factor. A famous 1906 result of Thue asserts that there exist arbitrarily long square-free words over a $3$-letter alphabet. We study square-free words with additional properties involving single-letter deletions and extensions of words. A square-free word is steady if it remains square-free after deletion of any single letter. We prove that there exist infinitely many steady words over a $4$-letter alphabet. We also demonstrate that one may construct steady words of any length by picking letters from arbitrary alphabets of size $7$ assigned to the positions of the constructed word. We conjecture that both bounds can be lowered to $4$, which is best possible. In the opposite direction, we consider square-free words that remain square-free after insertion of a single (suitably chosen) letter at every possible position in the word. We call them bifurcate. We prove a somewhat surprising fact, that over a fixed alphabet with at least three letters, every steady word is bifurcate. We also consider families of bifurcate words possessing a natural tree structure. In particular, we prove that there exists an infinite tree of doubly infinite bifurcate words over alphabet of size $12$.Comment: 11 pages, 1 figur

Bounds for the generalized repetition threshold

AbstractThe notion of the repetition threshold, which is the object of Dejean’s conjecture (1972), was generalized by Ilie et al. (2005) [8] to include the lengths of the avoided words. We give a lower and an upper bound on this generalized repetition threshold

A generalization of Thue freeness for partial words

This paper approaches the combinatorial problem of Thue freeness for partial words. Partial words are sequences over a finite alphabet that may contain a number of ?holes?. First, we give an infinite word over a three-letter alphabet which avoids squares of length greater than two even after we replace an infinite number of positions with holes. Then, we give an infinite word over an eight-letter alphabet that avoids longer squares even after an arbitrary selection of its positions are replaced with holes, and show that the alphabet size is optimal. We find similar results for overlap-free partial words

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

Combinatorial aspects of root lattices and words

This thesis is concerned with two topics that are of interest for the theory of aperiodic order. In the first part, the similar sublattices and coincidence site lattices of the root lattice A4 are analysed by means of a particular quaternion algebra. Dirichlet series generating functions are derived, which count the number of similar sublattices, respectively coincidence site lattices, of each index. In the second part, several strategies to derive upper and lower bounds for the entropy of certain sets of powerfree words are presented. In particular, Kolpakov's arguments for the derivation of lower bounds for the entropy of powerfree words are generalised. For several explicit sets we derive very good upper and lower bounds for their entropy. Notably, Kolpakov's lower bounds for the entropy of ternary squarefree, binary cubefree and ternary minimally repetitive words are confirmed exactly

Theoretical and Practical Aspects Related to the Avoidability of Patterns in Words

This thesis concerns repetitive structures in words. More precisely, it contributes to studying appearance and absence of such repetitions in words. In the first and major part of this thesis, we study avoidability of unary patterns with permutations. The second part of this thesis deals with modeling and solving several avoidability problems as constraint satisfaction problems, using the framework of MiniZinc. Solving avoidability problems like the one mentioned in the past paragraph required, the construction, via a computer program, of a very long word that does not contain any word that matches a given pattern. This gave us the idea of using SAT solvers. Representing the problem-based SAT solvers seemed to be a standardised, and usually very optimised approach to formulate and solve the well-known avoidability problems like avoidability of formulas with reversal and avoidability of patterns in the abelian sense too. The final part is concerned with a variation on a classical avoidance problem from combinatorics on words. Considering the concatenation of i different factors of the word w, pexp_i(w) is the supremum of powers that can be constructed by concatenation of such factors, and RTi(k) is then the infimum of pexp_i(w). Again, by checking infinite ternary words that satisfy some properties, we calculate the value RT_i(3) for even and odd values of i