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

Różnorodne oblicza ciągów bez powtórzeń

Słynne twierdzenia Thuego zapewniają o istnieniu dowolnie długich słów bez kwadratów nad alfabetem trzyliterowym oraz dowolnie długich slow bez nakładek nad alfabetem dwuliterowym. Rozwalamy dwie konsekwencje jego badań: gry bez powtórzeń oraz bezkwadratowe kolorowania układów linii prostych. Większa cześć pracy jest poświęcona wariantowi rezultatu Thuego wywodzącemu się z teorii gier, gdzie słowo wynikowe jest konstruowane wspólnie przez dwoje graczy, którzy naprzemiennie dołączają litery na koniec istniejącego słowa. Jeden z graczy (Ania) dba o unikanie predefiniowanych powtórzeń, kiedy drugi z nich (Benek) próbuje je wymusić. Naszym celem jest charakteryzacja strategii wygrywających dla Ani zależnie od wielkości używanego alfabetu i rodzaju powtórzeń. W szczególności dostarczamy formalne algorytmy do unikania: nietrywialnych kwadratów nad alfabetem ośmioliterowym, nakładek nad alfabetem czteroliterowym oraz piątych potęg nad alfabetem binarnym. W pozostałej części pracy studiujemy następujące geometryczne aspekty zagadnienia Thuego. Mając dany zbiór L prostych na płaszczyźnie i zbiór P wszystkich punktów przecięcia prostych z L, jaka jest najmniejsza liczba kolorów potrzebna do pokolorowania P tak, że każda prosta w L jest bezkwadratowa? Jaka jest najmniejsza liczba kolorów potrzebna do pokolorowania płaszczyzny tak, że każda ścieżka grafu o krawędziach długości jednostkowej składającego się ze współliniowych punktów jest bezkwadratowa? Dowodzimy, ze ograniczenia górne na te liczby wynoszą odpowiednio: 405 oraz 36.The famous Thue theorems assert that there exist arbitrarily long words without squares over a 3-letter alphabet and arbitrarily long words without overlaps over a 2-letter alphabet. We consider two consequences of his researches: nonrepetitive games and squarefree colourings of line arrangements. The bigger p a rt of the thesis is devoted to a game-theoretic variant of Thue result, where a word is constructed jointly by two players who alternately append letters to the end of an existing word. One of the players (Ann) takes care of avoiding predefined repetitions, while the other one (Ben) tries to force them. Our aim is to characterize the winning strategies for Ann dependent on the size of the alphabet and the kind of the repetitions. In particular, we provide explicit algorithms for avoiding: non-trivial squares over an 8-letter alphabet, overlaps over a 4-letter alphabet, and 5th powers over a binary alphabet. In the remaining p a rt of the thesis we study the following geometric aspects of Thue problem. Given a set L of lines in the plane and a set P of all intersection points of lines in L, what is the least number of colours needed to colour P so that every line in L is squarefree? What is the least number of colours needed to colour the plane so that every p a th of the unit distance graph whose vertices are colinear is squarefree? We prove that upper bounds for these numbers are respectively: 405 and 36

Overlap-Free Words and Generalizations

The study of combinatorics on words dates back at least to the beginning of the 20th century and the work of Axel Thue. Thue was the first to give an example of an infinite word over a three letter alphabet that contains no squares (identical adjacent blocks) xx. This result was eventually used to solve some longstanding open problems in algebra and has remarkable connections to other areas of mathematics and computer science as well. This thesis will consider several different generalizations of Thue's work. In particular we shall study the properties of infinite words avoiding various types of repetitions. In Chapter 1 we introduce the theory of combinatorics on words. We present the basic definitions and give an historical survey of the area. In Chapter 2 we consider the work of Thue in more detail. We present various well-known properties of the Thue-Morse word and give some generalizations. We examine Fife's characterization of the infinite overlap-free words and give a simpler proof of this result. We also present some applications to transcendental number theory, generalizing a classical result of Mahler. In Chapter 3 we generalize a result of Seebold by showing that the only infinite 7/3-power-free binary words that can be obtained by iterating a morphism are the Thue-Morse word and its complement. In Chapter 4 we continue our study of overlap-free and 7/3-power-free words. We discuss the squares that can appear as subwords of these words. We also show that it is possible to construct infinite 7/3-power-free binary words containing infinitely many overlaps. In Chapter 5 we consider certain questions of language theory. In particular, we examine the context-freeness of the set of words containing overlaps. We show that over a three-letter alphabet, this set is not context-free, and over a two-letter alphabet, we show that this set cannot be unambiguously context-free. In Chapter 6 we construct infinite words over a four-letter alphabet that avoid squares in any arithmetic progression of odd difference. Our constructions are based on properties of the paperfolding words. We use these infinite words to construct non-repetitive tilings of the integer lattice. In Chapter 7 we consider approximate squares rather than squares. We give constructions of infinite words that avoid such approximate squares. In Chapter 8 we conclude the work and present some open problems

Repetition in Words

The main topic of this thesis is combinatorics on words. The field of combinatorics on words dates back at least to the beginning of the 20th century when Axel Thue constructed an infinite squarefree sequence over a ternary alphabet. From this celebrated result also emerged the subfield of repetition in words which is the main focus of this thesis. One basic tool in the study of repetition in words is the iteration of morphisms. In Chapter 1, we introduce this tool among other basic notions. In Chapter 2, we see applications of iterated morphisms in several examples. The second half of the chapter contains a survey of results concerning Dejean's conjecture. In Chapter 3, we generalize Dejean's conjecture to circular factors. We see several applications of iterated morphism in this chapter. We continue our study of repetition in words in Chapter 4, where we study the length of the shortest repetition-free word in regular languages. Finally, in Chapter 5, we conclude by presenting a number of open problems

Periodicity, repetitions, and orbits of an automatic sequence

We revisit a technique of S. Lehr on automata and use it to prove old and new results in a simple way. We give a very simple proof of the 1986 theorem of Honkala that it is decidable whether a given k-automatic sequence is ultimately periodic. We prove that it is decidable whether a given k-automatic sequence is overlap-free (or squareefree, or cubefree, etc.) We prove that the lexicographically least sequence in the orbit closure of a k-automatic sequence is k-automatic, and use this last result to show that several related quantities, such as the critical exponent, irrationality measure, and recurrence quotient for Sturmian words with slope alpha, have automatic continued fraction expansions if alpha does.Comment: preliminary versio