Privileged Words and Sturmian Words

This dissertation has two almost unrelated themes: privileged words and Sturmian words. Privileged words are a new class of words introduced recently. A word is privileged if it is a complete first return to a shorter privileged word, the shortest privileged words being letters and the empty word. Here we give and prove almost all results on privileged words known to date. On the other hand, the study of Sturmian words is a well-established topic in combinatorics on words. In this dissertation, we focus on questions concerning repetitions in Sturmian words, reproving old results and giving new ones, and on establishing completely new research directions. The study of privileged words presented in this dissertation aims to derive their basic properties and to answer basic questions regarding them. We explore a connection between privileged words and palindromes and seek out answers to questions on context-freeness, computability, and enumeration. It turns out that the language of privileged words is not context-free, but privileged words are recognizable by a linear-time algorithm. A lower bound on the number of binary privileged words of given length is proven. The main interest, however, lies in the privileged complexity functions of the Thue-Morse word and Sturmian words. We derive recurrences for computing the privileged complexity function of the Thue-Morse word, and we prove that Sturmian words are characterized by their privileged complexity function. As a slightly separate topic, we give an overview of a certain method of automated theorem-proving and show how it can be applied to study privileged factors of automatic words. The second part of this dissertation is devoted to Sturmian words. We extensively exploit the interpretation of Sturmian words as irrational rotation words. The essential tools are continued fractions and elementary, but powerful, results of Diophantine approximation theory. With these tools at our disposal, we reprove old results on powers occurring in Sturmian words with emphasis on the fractional index of a Sturmian word. Further, we consider abelian powers and abelian repetitions and characterize the maximum exponents of abelian powers with given period occurring in a Sturmian word in terms of the continued fraction expansion of its slope. We define the notion of abelian critical exponent for Sturmian words and explore its connection to the Lagrange spectrum of irrational numbers. The results obtained are often specialized for the Fibonacci word; for instance, we show that the minimum abelian period of a factor of the Fibonacci word is a Fibonacci number. In addition, we propose a completely new research topic: the square root map. We prove that the square root map preserves the language of any Sturmian word. Moreover, we construct a family of non-Sturmian optimal squareful words whose language the square root map also preserves.This construction yields examples of aperiodic infinite words whose square roots are periodic.Siirretty Doriast

Critical Exponents and Stabilizers of Infinite Words

This thesis concerns infinite words over finite alphabets. It contributes to two topics in this area: critical exponents and stabilizers. Let w be a right-infinite word defined over a finite alphabet. The critical exponent of w is the supremum of the set of exponents r such that w contains an r-power as a subword. Most of the thesis (Chapters 3 through 7) is devoted to critical exponents. Chapter 3 is a survey of previous research on critical exponents and repetitions in morphic words. In Chapter 4 we prove that every real number greater than 1 is the critical exponent of some right-infinite word over some finite alphabet. Our proof is constructive. In Chapter 5 we characterize critical exponents of pure morphic words generated by uniform binary morphisms. We also give an explicit formula to compute these critical exponents, based on a well-defined prefix of the infinite word. In Chapter 6 we generalize our results to pure morphic words generated by non-erasing morphisms over any finite alphabet. We prove that critical exponents of such words are algebraic, of a degree bounded by the alphabet size. Under certain conditions, our proof implies an algorithm for computing the critical exponent. We demonstrate our method by computing the critical exponent of some families of infinite words. In particular, in Chapter 7 we compute the critical exponent of the Arshon word of order n for n ≥ 3. The stabilizer of an infinite word w defined over a finite alphabet Σ is the set of morphisms f: Σ*→Σ* that fix w. In Chapter 8 we study various problems related to stabilizers and their generators. We show that over a binary alphabet, there exist stabilizers with at least n generators for all n. Over a ternary alphabet, the monoid of morphisms generating a given infinite word by iteration can be infinitely generated, even when the word is generated by iterating an invertible primitive morphism. Stabilizers of strict epistandard words are cyclic when non-trivial, while stabilizers of ultimately strict epistandard words are always non-trivial. For this latter family of words, we give a characterization of stabilizer elements. We conclude with a list of open problems, including a new problem that has not been addressed yet: the D0L repetition threshold

디오판틴 근사, 연분수, 동역학적 스펙트럼

학위논문(박사) -- 서울대학교대학원 : 자연과학대학 수리과학부, 2023. 2. 임선희.Diophantine approximation is a rational approximation to an irrational number, which has been investigated using continued fractions. In the thesis, we deal with three topics related to Diophantine approximation and continued fractions. The first topic is the Markoff and Lagrange spectrum associated with the Hecke group. The classical Markoff and Lagrange spectrum is associated with the modular group PSL(2,$\mathbb Z$)=H_3, which has been studied using the regular continued fraction. We consider the Markoff and Lagrange spectrum associated with H_4 and H_6. We use the Romik dynamical system to show that some results on the classical Markoff and Lagrange spectra appear in the Markoff and Lagrange spectra associated with the Hecke group. The second topic is the exponents of repetition of Sturmian words. The exponent of repetition of a Sturmian word gives the irrationality exponent of the Sturmian number associated with the Sturmian word. For an irrational number $\theta$, we determine the minimum of the exponents of repetition of Sturmian words of slope $\theta$. We also investigate the spectrum of the exponents of repetition of Sturmian words of the golden ratio. The last topic is quasi-Sturmian colorings on regular trees. We characterize quasi-Sturmian colorings of regular trees by its quotient graph and its recurrence function. We obtain an induction algorithm of quasi-Sturmian colorings which is analogous to the continued fraction algorithm of Sturmian words.디오판틴 근사는 무리수의 유리수 근사를 뜻하는데 연분수를 사용하여 연구되어 왔습니다. 이 논문에서는 디오판틴 근사와 연분수에 관련된 세 가지 주제를 다루고 있습니다. 첫 번째 주제는 헤케군에 관련된 마르코프와 라그랑지 스펙트럼입니다. 고전적인 마르코프와 라그랑지 스펙트럼은 모듈러군 PSL(2,$\mathbb Z$)=H_3와 관련이 있는데, 단순연분수를 사용하여 연구되어 왔습니다. 우리는 H_4 와 H_6에 관련된 마르코프와 라그랑지 스펙트럼을 다룹니다. 우리는 로믹 동역학을 이용하여 고전적인 마르코프와 라그랑지 스펙트럼에서 발견된 결과가 헤케군에 관련된 마르코프와 라그랑지 스펙트럼에서도 나타남을 보입니다. 두 번째 주제는 스터미안 단어의 반복지수입니다. 스터미안 단어의 반복지수는 그 스터미안 단어와 연관된 스터미안 수의 비합리성 지수를 줍니다. 주어진 무리수 $\theta$에 대해, 우리는 기울기가 $\theta$인 스터미안 단어의 반복지수 중 최소값을 밝힙니다. 또한 우리는 황금비를 기울기로 갖는 스터미안 단어의 반복지수들의 스펙트럼을 연구합니다. 마지막 주제는 정규나무 위에서의 준-스터미안 채색입니다. 우리는 정규나무의 준-스터미안 채색을 이것의 몫 그래프와 회귀함수로 구분짓습니다. 우리는 스터미안 단어의 연분수 알고리즘과 유사한 준-스터미안 채색의 귀납적 알고리즘을 밝힙니다.1 Introduction 1 2 Diophantine approximation 7 2.1 Continued fraction 8 2.1.1 Basic properties 8 2.1.2 Gauss map 10 2.2 The Markoff and Lagrange spectra 11 3 The Markoff and Lagrange spectra associated with the Hecke group 16 3.1 The Markoff and Lagrange spectra on H_4 17 3.1.1 The Markoff and Lagrange spectra of the index 2 sublattice 17 3.1.2 The Markoff spectrum and the Romik expansion 23 3.1.3 Closedness of the Markoff spectrum 33 3.1.4 Hausdorff dimension of the Lagrange spectrum 34 3.1.5 Gaps of the Markoff spectrum 37 3.1.6 Halls Ray 40 3.2 The Markoff and Lagrange spectra on H_6 48 3.2.1 The Markoff spectrum and the Romik expansion 48 3.2.2 Closedness of the Markoff spectrum 53 3.2.3 Hausdorff dimension of the Lagrange spectrum 54 3.2.4 Gaps of the Markoff spectrum 57 4 Combinatorics on words 68 4.1 Sturmian words 68 4.2 The exponent of repetition 71 5 The spectrum of the exponents of repetition 74 5.1 The exponents of repetition of Sturmian words 74 5.2 The spectrum of the exponents of repetition of Fibonacci words 87 6 Colorings of regular trees 98 6.1 Sturmian colorings of trees 98 6.2 Quasi-Sturmian colorings 101 6.2.1 Quotient graphs of quasi-Sturmian colorings 102 6.2.2 Evolution of factor graphs 105 6.2.3 Quasi-Sturmian colorings of bounded type 110 6.2.4 Recurrence functions of colorings of trees 112 Bibliography 118 Abstract (in Korean) 122 Acknowledgement (in Korean) 123박

On k-abelian equivalence and generalized Lagrange spectra

We study the set of $k$-abelian critical exponents of all Sturmian words. It has been proven that in the case $k = 1$ this set coincides with the Lagrange spectrum. Thus the sets obtained when $k > 1$ can be viewed as generalized Lagrange spectra. We characterize these generalized spectra in terms of the usual Lagrange spectrum and prove that when $k > 1$ the spectrum is a dense non-closed set. This is in contrast with the case $k = 1$, where the spectrum is a closed set containing a discrete part and a half-line. We describe explicitly the least accumulation points of the generalized spectra. Our geometric approach allows the study of $k$-abelian powers in Sturmian words by means of continued fractions.</p

Subword complexity and power avoidance

We begin a systematic study of the relations between subword complexity of infinite words and their power avoidance. Among other things, we show that -- the Thue-Morse word has the minimum possible subword complexity over all overlap-free binary words and all $(\frac 73)$-power-free binary words, but not over all $(\frac 73)^+$-power-free binary words; -- the twisted Thue-Morse word has the maximum possible subword complexity over all overlap-free binary words, but no word has the maximum subword complexity over all $(\frac 73)$-power-free binary words; -- if some word attains the minimum possible subword complexity over all square-free ternary words, then one such word is the ternary Thue word; -- the recently constructed 1-2-bonacci word has the minimum possible subword complexity over all \textit{symmetric} square-free ternary words.Comment: 29 pages. Submitted to TC

Decision Algorithms for Ostrowski-Automatic Sequences

We extend the notion of automatic sequences to a broader class, the Ostrowski-automatic sequences. We develop a procedure for computationally deciding certain combinatorial and enumeration questions about such sequences that can be expressed as predicates in first-order logic. In Chapter 1, we begin with topics and ideas that are preliminary to this work, including a small introduction to non-standard positional numeration systems and the relationship between words and automata. In Chapter 2, we define the theoretical foundations for recognizing addition in a generalized Ostrowski numeration system and formalize the general theory that develops our decision procedure. Next, in Chapter 3, we show how to implement these ideas in practice, and provide the implementation as an integration to the automatic theorem-proving software package -- Walnut. Further, we provide some applications of our work in Chapter 4. These applications span several topics in combinatorics on words, including repetitions, pattern-avoidance, critical exponents of special classes of words, properties of Lucas words, and so forth. Finally, we close with open problems on decidability and higher-order numeration systems and discuss future directions for research