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) <i>xx</i>. 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. In this thesis we primarily study several variations of the problems studied by Thue in his work on repetitions in words, including some recent connections to other areas, such as graph theory. In Chapter 1 we give a brief introduction to the subject of combinatorics on words. In Chapter 2 we use uniform morphisms to construct an infinite binary word that contains no cubes <i>xxx</i> and no squares <i>yy</i> with |<i>y</i>| &#8805; 4, thus giving a simpler construction than that of Dekking. We also use uniform morphisms to construct an infinite binary word avoiding all squares except 0??, 1??, and (01)??, thus giving a simpler construction than that of Fraenkel and Simpson. We give some new enumeration results for these avoidance properties and solve an open problem of Prodinger and Urbanek regarding the perfect shuffle of infinite binary words that avoid arbitrarily large squares. In Chapter 3 we examine ternary squarefree words in more detail, and in Chapter 4 we study words <i>w</i> satisfying the property that for any sufficiently long subword <i>w'</i> of <i>w</i>, <i>w</i> does not contain the reversal of <i>w'</i> as a subword. In Chapter 5 we discuss an application of the property of squarefreeness to colourings of graphs. In Chapter 6 we study strictly increasing sequences (<i>a</i>(<i>n</i>))<i>n</i>&#8805;0 of non-negative integers satisfying the equation <i>a</i>(<i>a</i>(<i>n</i>)) = <i>dn</i>. Finally, in Chapter 7 we give a brief conclusion and present some open problems.

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.

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) <i>xx</i>. 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. In this thesis we primarily study several variations of the problems studied by Thue in his work on repetitions in words, including some recent connections to other areas, such as graph theory. In Chapter 1 we give a brief introduction to the subject of combinatorics on words. In Chapter 2 we use uniform morphisms to construct an infinite binary word that contains no cubes <i>xxx</i> and no squares <i>yy</i> with |<i>y</i>| &#8805; 4, thus giving a simpler construction than that of Dekking. We also use uniform morphisms to construct an infinite binary word avoiding all squares except 0², 1², and (01)², thus giving a simpler construction than that of Fraenkel and Simpson. We give some new enumeration results for these avoidance properties and solve an open problem of Prodinger and Urbanek regarding the perfect shuffle of infinite binary words that avoid arbitrarily large squares. In Chapter 3 we examine ternary squarefree words in more detail, and in Chapter 4 we study words <i>w</i> satisfying the property that for any sufficiently long subword <i>w'</i> of <i>w</i>, <i>w</i> does not contain the reversal of <i>w'</i> as a subword. In Chapter 5 we discuss an application of the property of squarefreeness to colourings of graphs. In Chapter 6 we study strictly increasing sequences (<i>a</i>(<i>n</i>))<i>n</i>&#8805;0 of non-negative integers satisfying the equation <i>a</i>(<i>a</i>(<i>n</i>)) = <i>dn</i>. Finally, in Chapter 7 we give a brief conclusion and present some open problems.

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.

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.

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.

In this thesis, we show that several natural questions about automatic sequences can be expressed as logical predicates and then decided mechanically. We extend known results in this area to broader classes of sequences (e.g., paperfolding words), introduce new operations that extend the space of possible queries, and show how to process the results. We begin with the fundamental concepts and problems related to automatic sequences, and the corresponding numeration systems. Building on that foundation, we discuss the general logical framework that formalizes the questions we can mechanically answer. We start with a first-order logical theory, and then extend it with additional predicates and operations. Then we explain a slightly different technique that works on a monadic second- order theory, but show that it is ultimately subsumed by an extension of the first-order theory. Next, we give two applications: critical exponent and paperfolding words. In the critical exponent example, we mechanically construct an automaton that describes a set of rational numbers related to a given automatic sequence. Then we give a polynomial-time algorithm to compute the supremum of this rational set, allowing us to compute the critical exponent and many similar quantities. In the paperfolding example, we extend our mechanical procedure to the paperfolding words, an uncountably infinite collection of infinite words. In the following chapter, we address abelian and additive problems on automatic sequences. We give an example of a natural predicate which is provably inexpressible in our first-order theory, and discuss alternate methods for solving abelian and additive problems on automatic sequences. We close with a chapter of open problems, drawn from the earlier chapters.

In this thesis, we show that several natural questions about automatic sequences can be expressed as logical predicates and then decided mechanically. We extend known results in this area to broader classes of sequences (e.g., paperfolding words), introduce new operations that extend the space of possible queries, and show how to process the results. We begin with the fundamental concepts and problems related to automatic sequences, and the corresponding numeration systems. Building on that foundation, we discuss the general logical framework that formalizes the questions we can mechanically answer. We start with a first-order logical theory, and then extend it with additional predicates and operations. Then we explain a slightly different technique that works on a monadic second- order theory, but show that it is ultimately subsumed by an extension of the first-order theory. Next, we give two applications: critical exponent and paperfolding words. In the critical exponent example, we mechanically construct an automaton that describes a set of rational numbers related to a given automatic sequence. Then we give a polynomial-time algorithm to compute the supremum of this rational set, allowing us to compute the critical exponent and many similar quantities. In the paperfolding example, we extend our mechanical procedure to the paperfolding words, an uncountably infinite collection of infinite words. In the following chapter, we address abelian and additive problems on automatic sequences. We give an example of a natural predicate which is provably inexpressible in our first-order theory, and discuss alternate methods for solving abelian and additive problems on automatic sequences. We close with a chapter of open problems, drawn from the earlier chapters.

Combinatorics on words explore words – often called strings in the com- puter science community, or monoids in mathematics – and their structural properties. One of the most studied question deals with repetitions which are a form of redundancy. The thesis focuses on estimating the maximum number of distinct squares in a string of length n. Our approach is to study the combinatorial properties of these overlapping structures, nested systems, and obtain insights into the intricate patterns that squares create. Determin- ing the maximum number of repetitions in a string is of interest in different fields such as biology and computer science. For example, the question arrises when one tries to bound the number of repetitions in a gene or in a computer file to be data compressed. Specific strings containing many repetitions are often of interest for additional combinatorial properties. After a brief review of earlier results and an introduction to the question of bounding the maxi- mum number of distinct squares, we present the combinatorial insights and techniques used to obtain the main result of the thesis: a strengthening of the universal upper bound obtained by Fraenkel and Simpson in 1998. / Thesis / Doctor of Philosophy (PhD)

A morphism $\sigma$ is \emph{(strongly) unambiguous} with respect to a word $\alpha$ if there is no other morphism $\tau$ that maps $\alpha$ to the same image as $\sigma$. Moreover, $\sigma$ is said to be \emph{weakly unambiguous} with respect to a word $\alpha$ if $\sigma$ is the only \emph{nonerasing} morphism that can map $\alpha$ to $\sigma(\alpha)$, i.\,e., there does not exist any other nonerasing morphism $\tau$ satisfying $\tau(\alpha) = \sigma(\alpha)$. In the first main part of the present thesis, we wish to characterise those words with respect to which there exists a weakly unambiguous \emph{length-increasing} morphism that maps a word to an image that is strictly longer than the word. Our main result is a compact characterisation that holds for all morphisms with ternary or larger target alphabets. We also comprehensively describe those words that have a weakly unambiguous length-increasing morphism with a unary target alphabet, but we have to leave the problem open for binary alphabets, where we can merely give some non-characteristic conditions. \par The second main part of the present thesis studies the question of whether, for any given word, there exists a strongly unambiguous \emph{1-uniform} morphism, i.\,e., a morphism that maps every letter in the word to an image of length $1$. This problem shows some connections to previous research on \emph{fixed points} of nontrivial morphisms, i.\,e., those words $\alpha$ for which there is a morphism $\phi$ satisfying $\phi(\alpha) = \alpha$ and, for a symbol $x$ in $\alpha$, $\phi(x) \neq x$. Therefore, we can expand our examination of the existence of unambiguous morphisms to a discussion of the question of whether we can reduce the number of different symbols in a word that is not a fixed point such that the resulting word is again not a fixed point. This problem is quite similar to the setting of Billaud's Conjecture, the correctness of which we prove for a special case.